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I N E A R L G E B R A STEPHEN H. FRIEDBERG ARNOLD J. INSEL LAWRENCE E. SPENCE L i s t o i Aj A'1 Al A* Al3 A' (AB) Bi@---Bk B(V) 0* Px C Q cond(/4) C'{R) Coc C(R) C([0,1]) Cx D det(A) ^i dim(V) eA ?A EA F / ( ^ ) F' f S y m b o l s the ij-th entry of the matrix A the inverse of the matrix A the pseudoinverse of the matrix A the adjoint of the matrix A the matrix A with row i and column j deleted the transpose of the matrix A the matrix A augmented by the matrix B the direct sum of matrices Bi through B^ the set of bilinear forms on V the dual basis of 0 the T-cyclic basis generated by x the field of complex numbers the iih Gerschgorin disk the condition number of the matrix A set of functions / on R with / ' ' ' continuous set of functions with derivatives of every order the vector space of continuous functions on R the vector space of continuous functions on [0,1] the T-cyclic subspace generated by x the derivative operator on C the determinant of the matrix A the Kronecker delta the dimension of V limm_0 0 ( / + A + 4r + + AS) the ith standard vector of F' the eigenspace of T corresponding to A a field the polynomial f(x) evaluated at the matrix A the set of n-tuples with entries in a field F page 9 page 100 page 414 page 331 page 210 page 17 page 161 page 320 page 422 page 120 page 526 page 7 page 296 page 469 page 21 page 130 page 18 page 331 page 525 page 131 page 232 page 89 page 47 page 312 page 43 page 264 page 6 page 565 page 8 /(T) the polynomial /(') evaluated at the operator T J-'iS, F) the set of functions from S to a field F H space of continuous complex functions on [0. 2TT] / or / the // x n identity matrix lv or I the identity operator on V K generalized eigenspace of T corresponding to A K,;, {./: (T^ 7 ~ n C o n t e n t s Preface IX 1 Vector Spaces 1 1.1 Introduction 1 1.2 Vector Spaces 6 1.3 Subspaces 16 1.4 Linear Combinations and Systems of Linear Equations . . . . 24 1.5 Linear Dependence and Linear Independence 35 1.6 Bases and Dimension 42 1.7* Maximal Linearly Independent Subsets 58 Index of Definitions 62 2 Linear Transformat ions and Matr ices 64 2.1 Linear Transformations. Null Spaces, and Ranges 64 2.2 The Matrix Representation of a Linear Transformation . . . 79 2.3 Composition of Linear Transformations and Matrix Multiplication 86 2.4 Invertibility and Isomorphisms 99 2.5 The Change of Coordinate Matrix 110 2.6' Dual Spaces 119 2.7* Homogeneous Linear Differential Equations with Constant Coefficients 127 Index of Definitions 145 3 Elementary Ma t r i x Operat ions and Systems o f Linear Equations 147 3.1 Elementary Matrix Operations and Elementary Matrices . . 147 Sections denoted by an asterisk are optional. vi Table of Contents 3.2 The Rank of a Matrix and Matrix Inverses 152 3.3 Systems of Linear Equations Theoretical Aspects 168 3.4 Systems of Linear Equations Computational Aspects . . . . 182 Index of Definitions 198 4 Determinants 199 4.1 Determinants of Order 2 199 4.2 Determinants of Order n 209 4.3 Properties of Determinants 222 4.4 Summary Important Facts about Determinants 232 4.5* A Characterization of the Determinant 238 Index of Definitions 244 5 Diagonal izat ion 245 5.1 Eigenvalues and Eigenvectors 245 5.2 Diagonalizability 261 5.3s Matrix Limits and Markov Chains 283 5.4 Invariant Subspaces and the Cayley Hamilton Theorem . . . 313 Index of Definitions 328 6 Inner Produc t Spaces 329 6.1 Inner Products and Norms 329 6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements 341 6.3 The Adjoint of a Linear Operator 357 6.4 Normal and Self-Adjoint. Operators 369 6.5 Unitary and Orthogonal Operators and Their Matrices . . . 379 6.6 Orthogonal Projections and the Spectral Theorem 398 6.7* The Singular Value Decomposition and the Pseudoinverse . . 405 6.8* Bilinear and Quadratic Forms 422 6.9* Einstein As Special Theory of Relativity 151 6.10* Conditioning and the Rayleigh Quotient 464 6.11* The Geometry of Orthogonal Operators 472 Index of Definitions 480 Table of Contents 7 Canonical Forms 482 7.1 The Jordan Canonical Form I 482 7.2 The Jordan Canonical Form II 497 7.3 The Minimal Polynomial 516 7.4* The Rational Canonical Form 524 Index of Definitions 548 Appendices 549 A Sets 549 B Functions 551 C Fields 552 D Complex Numbers 555 E Polynomials 561 Answers t o Selected Exercises 5 7 1 Index 589 -- _ . z P r e f a c e The language and concepts of matrix theory and, more generally, of linear algebra have come into widespread usage in the social and natural sciences, computer science, and statistics. In addition, linear algebra continues to be of great importance in modern treatments of geometry and analysis. The primary purpose of this fourth edition of Linear Algebra is to present a careful treatment of the principal topics of linear algebra and to illustrate the power of the subject through a variety of applications. Our major thrust emphasizes the symbiotic relationship between linear transformations and matrices. However, where appropriate, theorems are stated in the more gen-eral infinite-dimensional case. For example, this theory is applied to finding solutions to a homogeneous linear differential equation and the best approx-imation by a trigonometric polynomial to a continuous function. Although the only formal prerequisite for this book is a one-year course in calculus, it requires the mathematical sophistication of typical junior and senior mathematics majors. This book is especially suited for a. second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis. The book is organized to permit a number of different courses (ranging from three to eight semester hours in length) to be taught from it. The core material (vector spaces, linear transformations and matrices, systems of linear equations, determinants, diagonalization. and inner product spaces) is found in Chapters 1 through 5 and Sections 6.1 through 6.5. Chapters 6 and 7, on inner product spaces and canonical forms, are completely independent and may be studied in either order. In addition, throughout the book are applications to such areas as differential equations, economics, geometry, and physics. These applications are not central to the mathematical development, however, and may be excluded at the discretion of the instructor. We have attempted to make it possible for many of the important topics of linear algebra to be covered in a one-semester course. This goal has led us to develop the major topics with fewer preliminaries than in a traditional approach. (Our treatment of the Jordan canonical form, for instance, does not require any theory of polynomials.) The resulting economy permits us to cover the core material of the book (omitting many of the optional sections and a detailed discussion of determinants) in a one-semester four-hour course for students who have had some prior exposure to linear algebra. Chapter 1 of the book presents the basic theory of vector spaces: sub-spaces, linear combinations, linear dependence and independence, bases, and dimension. The chapter concludes with an optional section in which we prove x Preface that every infinite-dimensional vector space has a basis. Linear transformations and their relationship to matrices are the subject of Chapter 2. We discuss the null space and range of a linear transformation, matrix representations of a linear transformation, isomorphisms, and change of coordinates. Optional sections on dual spaces and homogeneous linear differential equations end the chapter. The application of vector space theory and linear transformations to sys-tems of linear equations is found in Chapter 3. We have chosen to defer this important subject so that it can be presented as a consequence of the pre-ceding material. This approach allows the familiar topic of linear systems to illuminate the abstract theory and permits us to avoid messy matrix computa-tions in the presentation of Chapters 1 and 2. There arc occasional examples in these chapters, however, where we solve systems of linear equations. (Of course, these examples are not a part of the theoretical development.) The necessary background is contained in Section 1.4. Determinants, the subject of Chapter 4, are of much less importance than they once were. In a short course (less than one year), we prefer to treat determinants lightly so that more time may be devoted to the material in Chapters 5 through 7. Consequently we have presented two alternatives in Chapter 4 -a complete development of the theory (Sections 4.1 through 4.3) and a summary of important facts that are needed for the remaining chapters (Section 4.4). Optional Section 4.5 presents an axiomatic development of the determinant. Chapter 5 discusses eigenvalues, eigenvectors, and diagonalization. One of the most important applications of this material occurs in computing matrix limits. We have therefore included an optional section on matrix limits and Markov chains in this chapter even though the most general statement of some of the results requires a knowledge of the Jordan canonical form. Section 5.4 contains material on invariant subspaces and the Cayley Hamilton theorem. Inner product spaces are the subject of Chapter 6. The basic mathe-matical theory (inner products; the Grain Schmidt process: orthogonal com-plements; the adjoint of an operator; normal, self-adjoint, orthogonal and unitary operators; orthogonal projections; and the spectral theorem) is con-tained in Sections 6.1 through 6.6. Sections 6.7 through 6.11 contain diverse applications of the rich inner product space structure. Canonical forms are treated in Chapter 7. Sections 7.1 and 7.2 develop the Jordan canonical form, Section 7.3 presents the minimal polynomial, and Section 7.4 discusses the rational canonical form. There are five appendices. The first four, which discuss sets, functions, fields, and complex numbers, respectively, are intended to review basic ideas used throughout the book. Appendix E on polynomials is used primarily in Chapters 5 and 7, especially in Section 7.4. We prefer to cite particular results from the appendices as needed rather than to discuss the appendices Preface xi independently. The following diagram illustrates the dependencies among the various chapters. Chapter 1 Chapter 2 Chapter 3 Sections 4.1 4.3 or Section 4.4 Sections 5.1 and 5.2 Section 5.4 Chapter 6 Chapter 7 One final word is required about our notation. Sections and subsections labeled with an asterisk (*) are optional and may be omitted as the instructor sees fit. An exercise accompanied by the dagger symbol (f) is not optional, however -we use this symbol to identify an exercise that is cited in some later section that is not optional. DIFFERENCES BETWEEN THE THIRD AND FOURTH EDITIONS The principal content change of this fourth edition is the inclusion of a new section (Section 6.7) discussing the singular value decomposition and the pseudoinverse of a matrix or a linear transformation between finite-dimensional inner product spaces. Our approach is to treat this material as a generalization of our characterization of normal and self-adjoint operators. The organization of the text is essentially the same as in the third edition. Nevertheless, this edition contains many significant local changes that im-xii Preface prove the book. Section 5.1 (Eigenvalues and Eigenvectors) has been stream-lined, and some material previously in Section 5.1 has been moved to Sec-tion 2.5 (The Change of Coordinate Matrix). Further improvements include revised proofs of some theorems, additional examples, new exercises, and literally hundreds of minor editorial changes. We are especially indebted to Jane M. Day (San Jose State University) for her extensive and detailed comments on the fourth edition manuscript. Additional comments were provided by the following reviewers of the fourth edition manuscript: Thomas Banchoff (Brown University), Christopher Heil (Georgia Institute of Technology), and Thomas Shemanske (Dartmouth Col-lege). To find the latest information about this book, consult our web site on the World Wide Web. We encourage comments, which can be sent to us by e-mail or ordinary post. Our web site and e-mail addresses are listed below. web site: http://www.math.ilstu.edu/linalg e-mail: linalg@math.ilst u.edu Stephen H. Friedberg Arnold J. Insel Lawrence R. Spence 1 V e c t o r S p a c e s 1.1 Introduction 1.2 Vector Spaces 1.3 Subspaces 1.4 Linear Combinations and Systems of Linear Equations 1.5 Linear Dependence and Linear Independence 1.6 Bases and Dimension 1.7* Maximal Linearly Independent Subsets 1.1 INTRODUCTION Many familiar physical notions, such as forces, velocities,1 and accelerations, involve both a magnitude (the amount of the force, velocity, or acceleration) and a direction. Any such entity involving both magnitude and direction is called a 'vector.' A vector is represented by an arrow whose length denotes the magnitude of the vector and whose direction represents the direction of the vector. In most physical situations involving vectors, only the magnitude and direction of the vector are significant; consequently, we regard vectors with the same magnitude and direction as being equal irrespective of their positions. In tins section the geometry of vectors is discussed. This geometry is derived from physical experiments that, test the manner in which two vectors interact. Familiar situations suggest that when two like physical quantities act si-multaneously at a point, the magnitude of their effect need not equal the sum of the magnitudes of the original quantities. For example, a swimmer swim-ming upstream at the rate of 2 miles per hour against a current of 1 mile per hour does not progress at the rate of 3 miles per hour. Lor in this instance the motions of the swimmer and current, oppose each other, and the rate of progress of the swimmer is only 1 mile per hour upstream. If, however, the 'The word velocity is being used here in its scientific sense as an entity having both magnitude and direction. The magnitude of a velocity (without, regard for the direction of motion) is called its speed. 2 Chap. 1 Vector Spaces swimmer is moving downstream (with the current), then his or her rate of progress is 3 miles per hour downstream. Experiments show that if two like quantities act together, their effect is predictable. In this case, the vectors used to represent these quantities can be combined to form a resultant vector that represents the combined effects of the original quantities. This resultant vector is called the sum of the original vectors, and the rule for their combination is called the parallelogram law. (See Figure 1.1.) Figure 1.1 Parallelogram Law for Vector Addition. The sum of two vectors x and y that act at the same point P is the vector beginning at P that is represented by the diagonal of parallelogram having x and y as adjacent sides. Since opposite sides of a parallelogram are parallel and of equal length, the endpoint Q of the arrow representing x + y can also be obtained by allowing x to act at P and then allowing y to act at the endpoint of x. Similarly, the endpoint of the vector x + y can be obtained by first permitting y to act at P and then allowing x to act at the endpoint of y. Thus two vectors x and y that both act at the point P may be added 'tail-to-head'; that is, either x or y may be applied at P and a vector having the same magnitude and direction as the other may be applied to the endpoint of the first. If this is done, the endpoint of the second vector is the endpoint of x + y. The addition of vectors can be described algebraically with the use of analytic geometry. In the plane containing x and y, introduce a coordinate system with P at the origin. Let (o,i, (Z2) denote the endpoint of x and (fei, 62) denote the endpoint of y. Then as Figure 1.2(a) shows, the endpoint Q of x+y is (ay + b, a-2 + b?). Henceforth, when a reference is made to the coordinates of the endpoint. of a vector, the vector should be assumed to emanate from the origin. Moreover, since a vector- beginning at the origin is completely determined by its endpoint, we sometimes refer to the point x rather than the endpoint of the vector x if x is a vector emanating from the origin. Besides the operation of vector addition, there is another natural operation that, can be performed on vectorsthe length of a vector may be magnified Sec. 1.1 Introduction {taittaa) 1 (CM +6i,a2 4- hi) {ai +61,62] Figure 1.2 or contracted. This operation, called scalar multiplication, consists of mul-tiplying the vector by a real number. If the vector x is represented by an arrow, then for any nonzero real number t, the vector tx is represented by an arrow in the same direction if t > 0 and in the opposite direction if t < 0. The length of the arrow tx is |i| times the length of the arrow x. Two nonzero vectors x and y are called parallel if y = tx for some nonzero real number t. (Thus nonzero vectors having the same or opposite directions are parallel.) To describe scalar multiplication algebraically, again introduce a coordi-nate system into a plane containing the vector x so that x emanates from the origin. If the endpoint of x has coordinates (01,02), then the coordinates of the endpoint of tx are easily seen to be (toi,t02). (See Figure 1.2(b).) The algebraic descriptions of vector addition and scalar multiplication for vectors in a plane yield the following properties: 1. For all vectors x and y, x + y = y + x. 2. For all vectors x, y. and z, (x + y) + z = x + (y + z). 3. There exists a vector denoted 0 such that x + 0 = x for each vector x. 4. For each vector x, there is a vector y such that x + y = 0. 5. For each vector x, lx x. 6. For each pair of real numbers a and 6 and each vector x, (ab)x = a(bx). 7. For each real number a and each pair of vectors x and y, a(x + y) = ax + ay. 8. For each pair of real numbers a and b and each vector x, (a + b)x = ax + bx. Arguments similar to the preceding ones show that these eight properties, as well as the geometric interpretations of vector addition and scalar multipli-cation, are true also for vectors acting in space rather than in a plane. These results can be used to write equations of lines and planes in space. 4 Chap. 1 Vector Spaces Consider first the equation of a line in space that passes through two distinct points A and B. Let O denote the origin of a coordinate system in space, and let u and v denote the vectors that begin at. O and end at A and B, respectively. If w denotes the vector beginning at A and ending at B. then 'tail-to-head' addition shows that u + w v, and hence w v u, where u denotes the vector (l)u. (See Figure 1.3, in which the quadrilateral OABC is a parallelogram.) Since a scalar multiple of w is parallel to w but possibly of a different length than w, any point on the line joining A and B may be obtained as the endpoint of a vector that begins at A and has the form tw for some real number t. Conversely, the endpoint of every vector of the form tw that begins at A lies on the line joining A and B. Thus an equation of the line through A and B is x = u + tw = u + t(v u), where t is a real number and x denotes an arbitrary point on the line. Notice also that the endpoint C of the vector v u in Figure 1.3 has coordinates equal to the difference of the coordinates of B and A. ^C Figure 1.3 Example 1 Let A and B be points having coordinates (2,0,1) and (4,5,3), respectively. The endpoint C of the vector emanating from the origin and having the same direction as the vector beginning at A and terminating at B has coordinates (4,5,3) - ( -2 ,0 ,1) = (6, 5, 2). Hence the equation of the line through A and B is x = ( -2 ,0 .1)-R(6,5 ,2) . Now let A, IS, and C denote any three noncollinear points in space. These points determine a unique plane, and its equation can be found by use of our previous observations about vectors. Let u and v denote vectors beginning at A and ending at B and C, respectively. Observe that any point in the plane containing A, B, and C is the endpoint S of a vector x beginning at A and having the form su + tv for some real numbers s and t. The endpoint of su is the point of intersection of the line through A and B with the line through S Sec. 1.1 Introduction *-C Figure 1.4 parallel to the line through A and C. (See Figure 1.4.) A similar procedure locates the endpoint of tv. Moreover, for any real numbers 5 and i, the vector su + tv lies in the plane containing A, B, and C. It follows that an equation of the plane containing A, B, and C is x = A + su + tv. where s and t are arbitrary real numbers and x denotes an arbitrary point in the plane. Example 2 Let A, B, and C be the points having coordinates (1,0,2), (3,-2,4), and (1,8, -5), respectively. The endpoint of the vector emanating from the origin and having the same length and direction as the vector beginning at A and terminating at B is ( - 3 , - 2 , 4 ) - ( 1 , 0 , 2 ) = ( - 4 , - 2 , 2 ) . Similarly, the endpoint of a vector emanating from the origin and having the same length and direction as the vector beginning at A and terminating at C is (1,8, 5) (1,0,2) = (0,8,-7) . Hence the equation of the plane containing the three given points is x = (l,0,2) + .s( -4 , -2 ,2) + f.(0,8,-7). Any mathematical structure possessing the eight properties on page 3 is called a vector space. In the next section we formally define a vector space and consider many examples of vector spaces other than the ones mentioned above. 1. EXERCISES Determine whether the vectors emanating from the origin and termi-nating at the following pairs of points are parallel. Chap. 1 Vector Spaces (a) (3. 1,2) and (6,4,2) (b) ( -3 ,1 ,7) and (9 . -3 , -21 ) (c) (5 . -6 .7) and ( - 5 , 6 , - 7 ) (d) (2 .0 , -5) and (5,0,-2) 2. hind the equations of the lines through the following pairs of points in space. (a) (3 , -2 ,4) and (-5.7,1) (b) (2,4.0) and ( - 3 , - 6 . 0 ) (c) (3, 7. 2) and (3, 7. -8) (d) ( - 2 , - 1 , 5 ) and (3,9,7) 3. Find the equations of the planes containing the following points in space. (a) (2. - 5 , - 1 ) . (0,4.6). and ( -3 ,7 .1) (b) (3, - 6 , 7), ( -2 ,0 . -4) , and (5, -9 , -2) (c) ( -8 . 2,0), (1.3,0), and (6, -5 ,0) (d) (1.1.1). (5,5.5). and ( -6 ,4 ,2) 4. What arc the coordinates of the vector 0 in the Euclidean plane that satisfies property 3 on page 3? Justify your answer. 5. Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the point, with coordinates (01,02), then the vector tx that emanates from the origin terminates at the point with coord i n ates (/ a . Ia >) 6. Show that the midpoint of the line segment joining the points (a.b) and (c.d) is ((a c)/2,(6 + d)/2). 7. Prove that the diagonals of a parallelogram bisect each other. 1.2 VECTOR SPACES In Section 1.1. we saw that, with the natural definitions of vector addition and scalar' multiplication, the vectors in a plane; satisfy the eight properties listed on page 3. Many other familiar algebraic systems also permit definitions of addition and scalar multiplication that satisfy the same eight properties. In this section, we introduce some of these systems, but first we formally define this type of algebraic structure. Definitions. A vector space (or linear space) V over a field2 F consists of a set on which two operations (called addition and scalar mul-tiplication, respectively) are defined so that for each pair of elements x, y. 2Fields are discussed in Appendix C. Sec. 1.2 Vector Spaces 7 in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold. (VS 1) For all x, y in V, x + y = y + x (commutativity of addition). (VS 2) For all x, y, z in V, (x + y) + z = x + (y + z) (associativity of addition). (VS 3) There exists an element in V denoted by 0 such that x + 0 = x for each x in V. (VS 4) For each element x in V there exists an clement y in V such that x + y = 0. (VS 5) For each clement x in V, 1.x = x. (VS 6) For each pair of elements a, b in F and each element x in V, (ab)x a(bx). (VS 7) For each clement a in F and each pair of elements x, y in V, a(x + y) ax + ay. (VS 8) For each pair of elements o, b in F and each clement x in V, (a + b)x ax + bx. The elements x + y and ax are called the sum of x and y and the product of a and x, respectively. The elements of the field F are called scalars and the elements of the vector space V are called vectors. The reader should not confuse this use of the word 'vector' with the physical entity discussed in Section 1.1: the word 'vector' is now being used to describe any element of a vector space. A vector space is frequently discussed in the text without explicitly men-tioning its field of scalars. The reader is cautioned to remember, however, that every vector space is regarded as a vector space over a given field, which is denoted by F. Occasionally we restrict our attention to the fields of real and complex numbers, which are denoted R and C. respectively. Observe that (VS 2) permits us to define the addition of any finite number of vectors unambiguously (without the use of parentheses). In the remainder of this section we introduce several important examples of vector spaces that are studied throughout this text. Observe that in de-scribing a vector space, it is necessary to specify not only the vectors but also the operations of addition and scalar multiplication. An object of the form (ai ,02, , an) , where the entries a, a2 , . . . , a n are elements of a field F, is called an n-tuple with entries from F. The elements 8 Chap. 1 Vector Spaces 01,02 ' are called the entries or components of the //-tuple. Two n-tuples (,,) F', and c F, then ;/ I i'- (a.| I 61,02 + 62 ' +6) and CK = (cai,C02 Sec. 1.2 Vector Spaces 9 In this book, we denote matrices by capital italic letters (e.g., A. B, and C). and we denote the entry of a matrix A that lies in row i and column j by Ajj. In addition, if the number of rows and columns of a matrix are equal. the matrix is called square. Two m x n matrices A and B are called equal if all their corresponding entries are equal, that is, if Ajj = By for 1 < i < m and 1 < j < n. Example 2 The set of all mxn matrices with entries from a Held F is a vector space, which we denote by M m x n (F ) , with the following operations of matrix addition and scalar multiplication: For A, B C Mm x n (F) and c F, (4 + B)y = Ay + By and (cA)ij - Ci4y for 1 < i < m and 1 < j < n. For instance, 2 0 1 - 3 -r> 9 3 6 4 -- 3 1 - 2 5 1 3 and - 3 0 - 2 - 3 2 0 6 6 - 9 111 M2X3 R) Example 3 Let S be any nonempty set and /' be any field, and let F(S,F) denote the set of all functions from S to F. Two functions / and g in T(S, F) are called equal if f(s) = g(s) for each s S. The set J-'iS.F) is a vector space with the operations of addition and scalar multiplication defined for / . g G- .F(S. F) and c F by ( / + 0 )OO=/ (* )+0( ) and (c/)(.s)=c[/(,s)] for each s e S. Note that these are the familiar operations of addition and scalar multiplication for functions used in algebra and calculus. A polynomial with coefficients from a. field F is an expression of the form f(x) = anxn + an-1x1 + 10 Chap. 1 Vector Spaces otherwise, the degree of a, polynomial is defined to be the largest exponent of x that appears in the representation /(./;) = anxn + a n _i : r n - ] + + axx + a0 with a nonzero coefficient. Note that the polynomials of degree zero may be written in the form f(x) c for some nonzero scalar c. Two polynomials. f(x) = anxn + an-ixn~l + h axx + a0 and g(x) = bmxm + 6m_1x' '-1 + + blX + b0, are called equal if m = n and a-i = b,. for ? 0 .1 . . . . , n. When F is a field containing infinitely many scalars, we usually regard a polynomial with coefficients from F as a function from F into F. (See page 569.) In this case, the value of the function f(x) = anxn + an-yxn x + + ayx + a0 at c e F is the scalar /(c) = anc' + an_1cn l + + aye + aQ. Here either of the notations / or /(./;) is used for the polynomial function f{x) = anxn + o n _ i ^ n _ 1 + + axx + a0. Example 4 Let f(x) = anxn + an-ixn~l H h axx + a0 and g{x) = bmxm + 6m_iarm-1 + + 6,.r + 60 be polynomials with coefficients from a field F. Suppose that m < n, and define 6m_i = 6m +2 = = 6n = 0. Then g(x) can be written as 9{x) = bnx' + 6 n _ ix n _ 1 + + bix + b0. Define f(x)+g{x) - (on + 6)a;n + ( a n _ i+ 6n . i) .r'- ' + + (, + 6,).r + (a0 + 60) and for any c F, define c/(;r) = canxn + can. ixn~l H h coia; + ca0. With these operations of addition and scalar multiplication, the set of all polynomials with coefficients from F is a vector space, which we denote by P(F). Sec. 1.2 Vector Spaces 11 We will see in Exercise 23 of Section 2.4 that the vector space defined in the next, example is essentially the same as P(F). Example 5 Let F be any field. A sequence in F is a function a from the positive integers into F. In this book, the sequence o such that, o~(n) an for n 1 .2 . . . . is denoted {a,,}. Let V consist of all sequences {on} in F that have only a finite number of nonzero terms a. If {} and {6n} are in V and t F . define {/'.} + {6n} = {an + bn} and t{an} = {tan}. With these operations V is a vector space. Our next two examples contain sets on which addition and scalar multi-plication are defined, but which are not vector spaces. Example 6 Let S = {(ai, 02): 01,02 R}- For (CM , 0-2), (61,62) E S and c R, define (oi,a2) + (61,62) = (CM + 61,02 - 62) and c(oi,a2) = (cai,ca2). Since (VS f). (VS 2), and (VS 8) fail to hold, S is not a vector space with these operations. Example 7 Let S be as in Example 6. For (01,02), (61,62) S and c 6 R, define (a i. a-)) + (61,62) = [a 1 + b, 0) and c(o, a->) = (ca 1,0). Then S is not a vector space with these operations because (VS 3) (hence (VS 4)) and (VS 5) fail. We conclude this section with a few of the elementary consequences of the definition of a vector space. Theorem 1.1 (Cancellation Law for Vector Addition). If x, y, and z are vectors in a vector space V .such that x + z = y + z, then x = y. Proof. There exists a vector v in V such that 2 + v = 0 (VS 4). Thus x = x + 0 = x + (z + v) = (x + z) + v = (y + z) + v = y + (z + v) =y+0=y by (VS 2) and (VS 3). I Corollary 1. The vector 0 described in (VS 3) is unique. 12 Chap. 1 Vector Spaces Proof. Exercise. ]| Corollary 2. The vector y described in (VS 4) is unique. Proof. Exercise. i The vector 0 in (VS 3) is called the zero vector of V, and the vector y in (VS 4) (that is. the unique vector such that x + y = 0) is called the additive inverse of x and is denoted by x. The next result, contains some of the elementary properties of scalar mul-tiplication. Theorem 1.2. In any vector space V, the following statements are true: (a) Ox = 0 for each x 6 V. (b) (a)x = (ax) = a.(x) for each a F and each x V. (c) aO = 0 for each a F. Proof, (a) By (VS 8), (VS 3), and (VS 1), it follows that Ox + Ox - (0 + 0 > = 0:/: = 0.r +0 = 0 + Ox. Hence Ox = 0 by Theorem 1.1. (b) The vector (ax) is the unique element of V such that ax + [(ax)] 0. Thus if ax + (a)x 0, Corollary 2 to Theorem 1.1 implies that {-a).r = -(ax). But. by (VS 8), ax + (-a)x = [a + {-a)]x = Ox = 0 by (a). Consequently (a)x = (ax). In particular. (l)x = x. So, by (VS 6). a(-x)=a[(-l)x] = [a(-l)]x = (-a)x. The proof of (c) is similar to the proof of (a). 1 EXERCISES 1. Label the following statements as true or false. (a) Every vector space contains a zero vector. (b) A vector space may have more than one zero vector. (c) In any vector space, ax = bx implies that a = 6. (d) In any vector space, ax = ay implies that x = y. (e) A vector in F' may be regarded as a matrix in M.xi(F). (f) An m x n matrix has m columns and n rows. (g) In P(F). only polynomials of the same degree may be added. (h) If / and o are polynomials of degree n, then / + g is a polynomial of degree n. (i) If / is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n. Sec. 1.2 Vector Spaces 13 (j) A nonzero scalar of F may be considered to be a polynomial in P(F) having degree zero, (k) Two functions in F(S, F) are equal if and only if they have the same value at each element of S. 2. Write the zero vector of 3. If '3x4 (F). M = 1 2 3 4 5 6 what are Mia,M2i, and M22? 4. Perform the indicated operations. w 14 Chap. 1 Vector Spaces Downstream Crossings Fall Spring Summer Brook trout Rainbow trout Brown trout 9 * 1 1 (J 1 4 0 0 Record the upstream and downstream crossings in two 3 x 3 matrices. and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season. 6. At the end of May, a furniture1 store had the following inventory. Early Mediter-American Spanish ranean Danish Living room suites Bedroom suites Dining room suites 1 5 3 2 1 1 ] I 2 3 4 6 5 (i 1 3 2 0 1 1 3 2 ) 3 Record these data as a 3 x 4 matrix M. To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2M. If the inventory at the end of June is described by the matrix A = interpret 2M A. How many suites were sold during the June sale? 7. Let 5 = {(). 1} and F - R. In F(S, R), show that. / = g and / -t g = h. where f(t) = 2t + 1. g(t) = 1 + At - It2, and h(t) = 5' + 1. 8. In any vector space1 V, show that, (a + b)(x + y) = ax + ay + bx + by for any x.y V and any o,6 F. 9. Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). 10. Let V denote the set of all differentiablc real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3. Sec. 1.2 Vector Spaces 15 11. Let V = {()} consist of a single vector 0 and define 0 + 0 = 0 and cO = 0 for each scalar c in F. Prove that V is a vector space over F. (V is called the zero vector space.) 12. A real-valued function / defined on the real line is called an even func-tion if /(/,) = f(t) for each real number t. Prove that the set. of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space1. 13. Let V denote the set of ordered pairs of real numbers. If (01,02) and (6], 62) are elements of V and c R, define (a 1, 16 Chap. 1 Vector Spaces 19. Let V = {(01,02): 01,02 R}- Define addition of elements of V coor-dinatewise, and for (01,02) in V and c R, define ;o, 0) if c = 0 cai, 1 it c / 0. Is V a vector space over R with these operations? Justify your answer. 20. Let V be the set of sequences {an} of real numbers. (See Example 5 for the definition of a sequence.) For {}, {6n} V and any real number t. define {} + {6n} - {. + 6} and t{a} = {tan}. Prove that, with these operations. V is a vector space over R. 21. Let V and W be vector spaces over a field F. Let Z = {(v,w): v Vandv/; W}. Prove that Z is a vector space over F with the operations (vi,wi) + (v2.w2) = (ci + u2, w + w2) ;md c(v,Wi) = (cvi,cw1). 22. How many matrices arc there in the vector space Mm x n(Z2)? (See Appendix C.) 1.3 SUBSPACES In the study of any algebraic structure, it is of interest to examine subsets that possess the same structure as the set under consideration. The appropriate notion of substructure for vector spaces is introduced in this section. Definition. .4 subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V. In any vector space V, note that V and {0} are subspaces. The latter is called the zero subspace of V. Fortunately it is not. necessary to verify all of the vector space properties to prove that a subset is a subspace. Because properties (VS 1), (VS 2). (VS 5), (VS 6), (VS 7), and (VS 8) hold for all vectors in the vector space, these properties automatically hold for the vectors in any subset. Thus a subset W of a. vector space V is a subspace of V if and only if the following four properties hold. Sec. 1.3 Subspaces 17 1. x+y W whenever ./ c W and y ( W. (W is closed under addition.) 2. ex W whenever c /' and x C W. (W is closed under scalar multiplication.) 3. W has a zero vector. 4. Each vector in W has an additive inverse in W. The next theorem shows that the zero vector of W must be the same as the zero vector of V ami thai property I is redundant. Theorem 1.3. Let V be a vector space and W a subset ofV. Then W is a subspace ofV if and only if the following three conditions hold for the operations defined in V. (a) 0 e W . (b) x + y W whenever x i- W and y i W. (c) ex W whenever c - F and x C W. Proof. If W is a subspace of V. t hen W is a vector space wit h the operat ions of addition and scalar multiplication defined on V. Hence conditions (!>] and (c) hold, and there exists a vector 0' > W Mich that x 0' = X for each x W. But also x + 0 = x, and thus 0' - 0 by Theorem 1.1 (p. I I). So condition (a) holds. Conversely, if conditions (a), (b), and (c) hold, the discussion preceding tins theorem shows thai W is a subspace of V if the additive inverse of each vector in W lies in W. But if a i W. then ( -l)x C W by condition (c). and x = ( 1 )x by Theorem 1.2 (p. 12). Hence W is a subspace of V. I The preceding theorem provides a simple method for determining whether or not a given subset of a vector space is a subspace. Normally, it is this result that is used to prove that a subset is. in fact, a subspace. The transpose A1 of an m x n matrix A is the n x m matrix obtained from A by interchanging the rows with the columns: that is, (Al)ij = Aj%. For example. 1 - 2 3 0 5 - 1 2 3 1 2 2 3 A symmetric matrix is a matrix .4 such that .4' = A. For example, the 2 x 2 matrix displayed above is a symmetric matrix. Clearly, a symmetric matrix must be square. The set W of all symmetric matrices 'u :F) is a subspace of M n x (F) since the conditions of Theorem 1.3 hold: 1. I'he zero matrix is equal to its transpose and hence belongs to W. It is easily proved that for any matrices A and B and any scalars a and 6. (aA + bB)' = a A' + bB'. (See Exercise 3.) Using this fact, we show that the set of symmetric matrices is closed under addition and scalar multiplication. 18 Chap. 1 Vector Spaces 2. If A W and B W, then A' = A and B* = B. Thus (A + B)1 = A1 + Bf = A + B. so that A + B W. 3. If A W. then A' = A. So for any a F, we have1 (a/1)' = .4' = aA. Thus aA W. The example's that follow provide further illustrations of the concept of a subspace. The first, three are particularly important. Example 1 Let it be a nonnegative integer, and let P(F) consist of all polynomials in P(F) having ck'gree1 k'ss than or equal to n. Since the zc;ro polynomial has degree 1, it is in P.(F). Moreover, the1 sum of two polynomials with degrees less than or equal to n is another polynomial of degree less than or equal to n, and the product of a scalar and a polynomial of degree le'ss than or equal to n is a polynomial of degree less than or equal te> n. Se> P(F) is edoseel under addition and scalar multiplication. It therefore follows from Theorem 1.3 that P(F) is a subspace of P(F). Example 2 Lc1!. C(7?) denote the set of all continuous real-valued functions defined on R. Clearly C(7?) is a subset of the vector space1 T(R.R) defined in Example 3 of Section 1.2. We claim that C(/t*) is a subspace of J-(R.R). First note that the1 zero of 1F(R. II) is the1 constant function defined by f(t) = 0 for all t R. Since constant functions are1 continuous, we- have / C(R). Moreover, the sum of two continuous functions is continuous, and the product of a real number and a. continuous function is continuous. So C(R) is closed under addition and scalar multiplication and hene-e1 is a subspace of J-(R.R) by Theorem 1.3. Example 3 An n x n matrix M is called a. diagonal matrix if M,j = 0 whenever i ^ j. that is. if all its nondiagonal entries are zero. Clearly the1 zero matrix is a diagonal matrix because all of its emtric's are1 0. Moreover, if ,4 and B are diagonal v x n matrices, then whenever i ^ j, (A f B)ij = A,, + Bi:j = 0 + 0 = 0 and (cA)l:J = eA,, = cO = 0 for any scalar c. Hence1 A + B and cA are1 diagonal matrices for any scalar c. Therefore the set of diagonal matrices is a subspace of MX(F) by Theo-rem 1.3. Example 4 The1 trace of an n x n matrix M. denoted tr(M), is the- sum of the diagonal entries of M; that is. tr(M) = Mn + M 2 2 + --- + M n n . Sec. 1.3 Subspaces 19 It follows from Exercise 6 that the set of n x n matrices having trace equal to zero is a subspace of M n x n (F) . Example 5 The set of matrices in M,IIXII(R.) having nonnegative entries is not a subspace of Mmxn{R) because it is not closed under scalar multiplication (by negative scalars). The next theorem shows how te) form a new subspace from other sub-spaces. Theorem 1.4. Any intersection of subspaces of a vector si>ace V is a subspace ofV. Proof. Let C be a collection of subspaces of V. and lei W demote' the intersection of the1 subspaces in C. Since1 every subspace contains the zcre> vector. 0 W. Let c; /' and x,y W. Then x and y are' contained in each subspace in C. Because- each subspace in C is closed under addition and scalar multiplication, it follows that x + y and ax are contained in each subspace in C. Hence X f y and a.r are also contained in W, se> that W is a subspace of V by Theorem 1.3. I Having shown that the intersection of subspaces of a vector space V is a subspace of V. it is natural to consider whether or not the union of subspaces of V is a subspace of V. It is easily seen that the- union of subspaces must contain the zero vector and be' closed under scalar multiplication, but in general the union of subspaces of V need not be1 closed under addition. In fact, it can be readily shown that the union of two subspaces of V is a subspace of V if and only if one of the subspaces contains the other. (See Exercise 19.) There is. however, a natural way to combine two subspaces Wi and W_) to obtain a subspace that contains both W| and W-j. As we already have suggested, the key to finding such a subspace is to assure thai it must be closed under addition. This idea is explored in Exercise 23. EXERCISES 1. Label the following statements as true or false. (a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V. (b) The empty set is a subspace of every vector space. (c) If V is a vector space other than the zero vector space, then V contains a subspace W such thai W f V. (d) The intersection of any two subsets of V is a subspace of V. 20 Chap. 1 Vector Spaces (e) An n x n diagonal matrix can never have more than n nonzero entries. (f) The trace of a square matrix is the product of its diagonal entries. (g) Let W be the ay-plane in R3; that is, W = {(ai,a2 ,0): a i ,a 2 R}. ThenW = R2. 2. Determine the transpose of each of the matrices that follow. In addition, if the matrix is square, compute its trace. (b) (d) (v.) ( I 1 3 5) (f) (h) 0 8 - 6 3 4 7 - 2 5 I 4 7 0 1 - 6 3. Prove that (aA + bB)1 = aA* + bB1 for any A, B M m x n ( F ) and any a, 6 F. 4. Prove that (A1)* = A for each A M m X n (F) . 5. Prove that A + A1 is symmetric for any square matrix A. 6. Prove that tr(aA + bB) = atr(A) + 6tr() for any A. B M n x n (F) . 7. Prove that diagonal matrices are symmetric matrices. 8. Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3. Justify your answers. (a) W) = {(01,02,03) R3: ai = 3a2 and 03 = a2) (b) W2 - {(01,02,03) R3: a, = a3 + 2} (c) W3 = {(01,02,03) R3: 2ai - 7o2 + a3 = 0} (d) W4 = {(ai,a2 , a3) R3: O] - 4a2 - a3 = 0} (e) W5 = {(aj, a2,03) R3: ai + 2a2 - 3o3 = 1} (f) W6 = {(01, a2 , a3) R3: 5a? - 3a | + 60^ = 0} 9. Let Wi, W3, and VV4 be as in Exercise 8. Describe Wi n W3, Wi n W4, and W3 n W4, and observe that each is a subspace of R3. Sec. 1.3 Subspaces 21 10. Prove that W, = {(o i ,a 2 , . . . , a n ) F' : ai + o2 + + a n = 0} is a subspace of Fn, but W2 = {(a , a2,.... an) F n : a i + a2 H h an = 1} is not. 11. Is the set. W = {f(x) P(F): f(x) = 0 or f(x) has degree; n] a subspace of P(F) if n > 1? Justify your answer. 12. An m x n matrix A is called upper triangular if all entries lying below the diagonal entries are zero, that is, if Aij = 0 whenever i > j. Prove that, the upper triangular matrices form a subspace of M.m x n(F). 13. Let S be a nonempty set and F a. field. Prove that for any SQ S, {/ T(S, F): / (s 0 ) = 0}, is a subspace of F(S, F). 14. Let S be a nonempty set and F a. field. Let C(S, F) denote the set of all functions / T(S, F) such that f(s) 0 for all but a finite number of elements of S. Prove that C(S, F) is a subspace of ^(S, F). 15. Is the set of all differentiable real-valued functions defined on R a sub-space of C(R)? Justify your answer. 16. Let Cn(R.) denote the set of all real-valued functions defined on the real line that have a continuous nth derivative. Prove that Cn(R) is a subspace of F(R,R). 17. Prove that a subset W of a vector space V is a subspace; of V if and only if W ^ 0, and, whenever a F and x,y W, then ax W and x + y W. 18. Prove that a subset W of a vector space; V is a subspace of V if and only if 0 W and ax + y W whenever a F and x. y W. 19. Let W, and W2 be subspaces of a vector space V. Prove that W| U W2 is a subspace of V if and only if W, C W2 or W2 C W,. 20. ' Prove that if W is a subspace of a vector space V and w, w2, , wn are in W, then aW + a2w2 + + anwn W for any scalars a, a 2 , . . . , an. 21. Show that the set of convergent sequences {on} (i.e., those for which limn-too a,n exists) is a subspace of the vector space V in Exercise 20 of Section 1.2. 22. Let F and F2 be fiedds. A function g T(Fl,F2) is called an even function if g(t) = g(t) for each t F and is called an odd function if g(t) = g(t) for each t Fi. Prove that the set of all even functions in T(F, F2) and the set of all odd functions in T(I', F2) are subspace;s of.F(Fi,F2). 'A dagger means that, this exercise is essential for a later sectiejii. 22 Chap. 1 Vector Spaces The following definitions are used in Exercises 23 30. Definition. If S and S2 are nonempty subsets of a vector space V. then the sum of Si and S2, denoted S +S2, is the set [x+y: x S and y S2}. Definition. A vector space V is called the direct sum of W] and VV2 if W, and W2 are subspaces of'W such that W2 D W2 = {0} and W, +W 2 = V. Wc denote that V is the direct sum of W] and W2 by writing V = W, W2. 23. Let Wi and W2 be subspaces of a vector space V. (a) Prove that Wi + VV2 is a subspace of V that contains both W] and W2. (b) Prove that any subspace of V that contains both Wi and W2 must also contain W t + W2. 24. Show that. Fn is the direct sum of the subspaces W, = { ( o i , o 2 , . . . , o n ) F n : o n = 0} and W2 = {(01,o2 , . . . , on) F' : a , = a2 = = an_] = ()}. 25. Let W[ denote the set of all polynomials /(./:) in P(F) such that in the representation f(x) ~ anx' + fln_ia;'_1 -| h a ia + c/(l. we have o^ = 0 whenever / is even. Likewise let W2 denote the set of all polynomials g(x) in P(F) such that in the representation g(x) = bmx' + 6m_1a;m-1 + + bxx + b0, we have 6* = 0 whenever i is odd. Prove that P(F) = W, W2. 26. In M m x n ( F ) define W, - {A MTOXn(F): Ai3 = 0 whenever i > j} and W2 = {A M m X n (F ) : Ai;j - 0 whenever i < j}. (W, is the set of all upper triangular matrices defined in Exercise 12.) Show that M W X n ( F ) = W 1 ( D W 2 . 27. Let V demote the vector space consisting of all upper triangular n x n matrices (as defined in Exercise 12), and let W| denote the subspace of V consisting of all diagonal matrices. Show that V = Wi W2, where W2 = {A V: Aij = 0 whenever i > j). Sec. 1.3 Subspaces 23 28. A matrix M is called skew-symmetric if Ml = M. Clearly, a skew-symmetric matrix is square. Let F be a. fiedd. Prove that the set Wi of all skew-symmetric n x n matrices with entries from F is a subspae;e of M n x n (F) . Now assume that F is not of characteristic 2 (see Ap-pendix C), and let W2 be the subspace of M n x n ( F ) consisting of all symmetric n x n matrices. Prove that M n x n ( F ) Wi W2. 29. Let F be a field that is not of characteristic 2. Define W, - {A F): Aij = 0 whenever i < ?'} and W2 to be the set of all symmetric n x n matrices with entries from /'. Both Wi and W2 are subspaces of Mn X(F). Prove that M x n (F) = Wi + W2. Compare this exercise with Exercise 28. 30. Let Wi and W2 be subspaces of a vector space V. Prove that V is the direct sum of Wi and W2 if and only if each vector in V can be uniquely written as x. + x2: where ari Wi and x2 W2. 31. Let W be a subspace of a vector space V over a field F. For any v V the set {v} + W = {v + w: w W} is called the coset of W containing v. It is customary to denote this coset. by v + W rather than {v} + W. (a) Prove that v + W is a subspace of V if and only if v W. (b) Prove that c, + W = v2 + W if and only if vx - v2 W. Addition and scalar multiplication by scalars of F can be defined in the collection S = [v f W: v V} of all cosets of W as follows: (vi + W) + (v2 + W) = (Vj + v2) + W for all v,v2 V and for all v V and a F. a(v + W) = av + W (c) Prove that the preceding operations are well defined; that is, show that if vi + W = v[ + W and v2 + W = v'2 + W, then and (c, + W) + (v2 + W) = (v[ + W) + (v'2 + W) a(Vl + W) = a(v[ + W) (d) for all a F. Prove that the set S is a vector space with the operations defined in (c). This vector space is called the quotient space of V modulo W and is denoted by V/W. 24 Chap. 1 Vector Spaces 1.4 LINEAR COMBINATIONS AND SYSTEMS OF LINEAR EQUATIONS In Section 1.1. it was shown that the equaticju of the plane through three noncollinear points A, B, and C in space is x = A + su + tv, where u and v denote the vectors beginning at. A and ending at B and C, respectively, and s and t deneDte arbitrary real numbers. An important special case occurs when A is the origin. In this case, the equation of the plane simplifies to x = su + tv, and the set of all points in this plane is a subspace of R3. (This is proved as Theorem 1.5.) Expressions of the form su + tv, where s and t are scalars and u and v are vectors, play a central role in the theory of vector spacers. The appropriate; generalization of such expressions is presented in the following deli nit ions. Definitions. Let V be a vector space and S a nonempty subset ofV. A vector v V is called a linear combination of vectors of S if there exist a finite number of vectors u, u2,.... un in S and scalars ai. a 2 , . . . . an in F such that v = oiU + a2u2 + f anun. In this case we also say that v is a linear combination of ii, a2 ti and call a., a2...., an the coefficients of the linear combination. Observe that in any vector space V, Ov 0 for each v V. Thus the zero vector is a, linear combination of any nonempty subset of V. Example 1 TABLE 1.1 Vitamin Content of 100 Grams of Certain Foods A 13 [ I5_. Niacin C (units) (mg) (nig) (mg) (nig) Apple butter 0 0.01 0.02 0.2 2 Raw, unpared apples (freshly harvested) 90 0.03 0.02 0.1 4 Chocolate-coated candy with coconut 0 0.02 0.07 0.2 0 center Clams (meat, only) 100 0.10 0.18 L.3 10 Cupcake from mix (dry form) 0 0.05 0.06 0.3 0 Cooked farina (unenriched) (0)a 0.01 0.01 0.1 (0) Jams and preserves In 0.01 0.03 0.2 2 Coconut custard pie (baked from mix) 0 0.02 0.02 0.4 0 Raw brown rice (0) 0.34 0.05 4.7 (0) Soy sauce 0 0.02 0.25 0.4 0 Cooked spaghetti (unenriched) 0 0.01 0.01 0.3 0 Raw wild rice (0) 0.45 0.03 6.2 (0) Source: Rernice K. Watt and Annabel I.. Merrill, Composition of Foods (Agriculture Hand-book Number 8). Consumer and food Economics Research Division, U.S. Department of Agriculture, Washington, D.C., 1963. aZeros in parentheses indicate that the amount, of a vitamin present is either none or too small to measure. Sec. 1.4 Linear Combinations and Systems of Linear Equations 25 Table 1.1 shows the vitamin content of 100 grams of 12 foods with respect to vitamins A, Bi (thiamine), B2 (riboflavin), niacin, and C (ascorbic acid). The vitamin content of 100 grams of each food can be recorded as a column vector in R:> for example1, the vitamin vector for apple butter is /0.00 0.01 0.02 0.20 2.00J Considering the vitamin vectors for cupcake, coconut custard pie, raw brown rice, soy sauce, and wild rice1, we see that /0.0() 0.05 0.06 0.30 ^0.00y + /o.oo 0.02 0.02 0.40 lO.OOy 4 /0.00 0.34 0.05 4.70 Ki)i)oj + 2 /o.oo 0.02 0.25 0.40 0.00y = /0.00 0.45 0.63 6.20 0.()()/ Thus the vitamin vector for wild rice1 is a linear combination of the vitamin vectors for cupcake, coconut custard pie, raw brown rice, and soy sauce. So 100 grams of cupcake, 100 grains of coconut custard pie, 100 grams of raw brown rice, and 200 grains of soy sauce provide exactly the same amounts of the five vitamins as 100 grams of raw wild rice. Similarly, since fom 0.01 0.02 0.20 ^2.ooy + /90.00 0.03 0.02 0.10 v 4.00y -f /0.00 0.02 0.07 0.20 K0.00j + /o.oo 0.01 0.01 0.10 0.00yi + / io.oo 0.01 0.03 0.20 V 2-00y + /o.oo 0.01 0.01 0.30 yO.oo/ = /IOO.OOX 0.10 0.18 1.30 10.00/ 200 grains of apple butter. 100 grams of apples, 100 grams of chocolate candy, 100 grams of farina, 100 grams of jam, and 100 grams of spaghetti provide exactly the same amounts of the five vitamins as 100 grams of clams. Throughout. Chapters 1 and 2 we encounter many different situations in which it is necessary to determine1 whether or not a vector can be expressed as a linear combination of other vectors, and if so, how. This question often reduces to the problem e>f solving a system of linear equations. In Chapter 3, we discuss a general method for using matrices to solve any system of linear equations. For now. we illustrate how to solve a system of linear equations by showing how to determine if the vector (2,6,8) can be; expressed as a. linear combination of ui = (1. 2.1), u2 = ( -2 , -4 , -2) , u3 = (0,2,3), 26 Chap. 1 Vector Spaces u4 = (2,0, -3) , and u5 = (-3.8.16). Thus we must determine if there are scalars 01,02,03,04, and 05 such that (2,6,8) = tti'Ui + a2u2 + 03113 + ei4U,i + 05145 = Oi(l, 2,1) + a 2 ( -2 , - 4 , - 2 ) + o3(0,2,3) + a..,(2.0.-3) + a5(-3.8.10) = (ai 2o.2 + 24 3as, 2a 1 4a2 + 203 + 805, 2 03 2a,4 + 7a;3 = 1 (3) 04 2ar, = 3. Sec. 1.4 Linear Combinations and Systems of Linear Equations 27 We continue by eliminating a4 from every equation of (3) except the third. This yic'lds a i 2a2 + o,r> A o3 + 3 o 5 = 7 (4) 04 2a,r, = 3. System (4) is a system of the desired form: It is easy to solve for the first unknown present in each of the equations (01,03, and 04) in terms of the other unknowns (a2 and 0.5). Rewriting system (4) in this form, we find that Oi = 2a2 a.-5 4 03 = 3a-, + 7 a.4 = 2a 5 + 3. Thus for any choice of scalars a2 and a.5, a vector of the form (oj. a2.a-.iy 04,05) = (2a2 - 05 - 4. a2 , 3ar, + 7, 2a5 + 3,05) is a solution to system (1). In particular, the vector (4,0,7,3,0) obtained by setting a2 = 0 and a.5 = 0 is a solution to (1). Therefore (2,6,8) = -Aui + 0u2 + 7u3 + 3w4 + 0u5, so that. (2,6,8) is a linear combination of U,u2,1/3,04, and 1/5. The procedure: just illustrated uses three1 types of operations to simplify the original system: 1. interchanging the order of any two equations in the system; 2. multiplying any equation in the system by a nonzero constant; 3. adding a. constant, multiple of any equation to another equation in the system. In Section 3.4, we prove that these operations do not. change the set of solutions to the original system. Note that we employed these operations to obtain a system of equations that had the following properties: 1. The first nonzero coefficient in each equation is one. 2. If an unknown is the first, unknown with a. nonzero coefficient in some equation, then that, unknown occurs with a zero coefficient in each of the other equations. 3. The first unknown with a nonzero coefficient in any equation has a larger subscript than the first unknown with a nonzc:ro coefficient in any prex-cxling equation. 28 Chap. 1 Vector Spaces To help clarify the meaning of these properties, note that none of the following systems meets these requirements. X + 3X2 + X4 =7 ,_. 2 x 3 - 5 x 4 = - l ( 0 ) x,i - 2x2 + 3x3 + x5 = - 5 x3 - 2 x 5 = 9 (6) x4 + 3.T5 = 6 xi - 2x8 + xr} = 1 x4 - 6x5 = 0 (7) x2 + 5x3 - 3x5 = 2. Specifically, system (5) does not satisfy property 1 because; the first nonzero coefficient in the second equation is 2; system (6) does not. satisfy property 2 because X3, the first unknown with a nonzero ccxuficient in the second equa-tion, occurs with a nonzero e;oeffieie:nt in the first equation; and system (7) doe:s not. satisfy property 3 because x2, the first unknown with a nonzero coefficient, in the third equation, does not have a larger subscript than ./|. the first unknown with a nonzero coefficient, in the second equation. One:e: a system with properties 1, 2. and 3 has been obtained, it is easy to solve lor some of the unknowns in terms of the others (as in the preceding example). If, however, in the course of tisiug operations 1, 2, and 3 a system containing an equation of the form 0 = c, where c is nonzero, is obtained, then the original system has no solutions. (Sex1 Example 2.) We return to the study of systems of linear equations in Chapter 3. We diseniss there the theoretical basis for this method of solving systems of linear equations and further simplify the procedure by use of matrices. Example 2 We claim that 2x3 - 2x2 + 12x - 6 is a linear combination of x3 - 2x2 - 5x - 3 and 3x3 - 5x2 - 4x - 9 in P-.i(R), but that, 3x3 - 2x2 + 7x + 8 is not. In the first, case we wish to find scalars a and b such that 2x:j - 2x2 + 12x - 6 = a(xz - 2x2 - 5x - 3) + b(3x:i - -r>x2 - 4x - 9) Sec. 1.4 Linear Combinations and Systems of Linear Equations 29 = (a + 3b)x3 + ( -2o - 56)x2 + ( -5a - 46)x + (-3a. - 96). Thus we are led to the following system of linear equations: a + 36 = 2 - 2 a - 56 = - 2 - 5 a - 46 = 12 - 3 a - 96 = -6 . Adding appropriate multiples of the first equation to the others in order to eliminate a. we find that a + 36 = 2 6 = 2 116 = 22 06= 0. Now adding the appropriate multiples of the second equation to the others yields o = - 4 6 = 2 0 = 0 0 = 0. Hence 2x3 - 2x2 + 12x - 6 = -A(x3 - 2x2 - 5x - 3) + 2(3x:i - 5x2 - 4x - 9). In the second case, we wish to show that, there are no scalars a and 6 for which 3x3 - 2x2 + 7x + 8 = a(x3 - 2x2 - 5x - 3) + 6(3x3 - 5x2 - 4x - 9). Using the preceding technique, we obtain a system of linear equations a + 36 = 3 - 2 a - 56 = - 2 - 5 a - 46 = 7 - 3 a - 96 = 8. (8) Eliminating a. as before yields a + 36= 3 6 = 4 116 = 22 0 = 17. But the presence of the inconsistent equation 0 = 17 indicates that (8) has no solutions. Hence 3x3 2x2 + 7x + 8 is not a linear combination of x:i - 2x2 - 5x - 3 and 3x3 - 5x2 - 4x - 9. 30 Chap. 1 Vector Spaces Throughout this book, we form the set of all linear combinations of some set of vectors. We now name such a. set of linear combinations. Definition. Let S be a nonempty subset of a vector space V. The span of S, denoted span(S'), is the set consisting of all linear combinations of the vectors in S. For convenience, we define span(0) = {()}. In R3, for instance, the span of the set {(1.0,0), (0.1, 0)} consists of all vectors in R3 that have the form a( 1,0,0) 4- 6(0,1.0) = (a, 6,0) for some scalars a and 6. Thus the span of {(1,0,0), (0, 1,0)} contains all the points in the xy-plane. In this case, the span of the set is a subspace of R . This fact is true in general. Theorem 1.5. The span of any subset S of a vector space V is a subspace of V. Moreover, any subspace of V that contains S must also contain the span of S. Proof This result is immediate if .S1 = 0 because span(0) = {()}, which is a subspace that is contained in any subspace of V. If S ^ 0 , then S contains a vector z. So Oz = 0 is in span(5'). Let ^iV span (5). Then there exist vectors U, 2 , . . . , um, V.v2...., v in 5 and scalars oi, 02 , . . . , am, b, 6 2 , . . . , bn such that x = aii + a2u2 H f- amu.m and y = 6, c, + b2v2 - - bnvn. Then x + y = a, i + a2u.2 + + amum + 6|t,1 + 62t'2 4 + be and, for any scalar c, ex = (cai)wi + (c.a2)u2 4 f- (cam)um are clearly linear combinations of the vectors in S: so x + y and ex are in span(5). Thus span(S) is a subspace of V. Now let W denote any subspace of V that contains S. If w C span(.S'). then w has the form w = CiWj +C2if2 + ' -+CkWk for some vectors W, w2 u. in S and some scalars c, c2,..., c.f,-. Since S C W, we have u), w2,..., Wk W. Therefore w = c+wi + c2w2 + + Ck'Wk is in W by Exercise 20 of Section 1.3. Because iu, an arbitrary vector in span(Sec. 1.4 Linear Combinations and Systems of Linear Equations 31 Example 3 The vectors (1,1,0), (1,0,1), and (0,1,1) generate R3 since an arbitrary vector {0.1,0,2,0,3) in R3 is a linear combination of the three given vectors; in fact, the scalars r, s, and t for which r( l , 1,0) + a(l, 0,1) + t{0,1,1) = (aua2,a3) are r = -(a + 02 a3), s = -{a - a2 + a3), and t = -(a + 0,2 + a3). Example 4 The polynomials x2 + 3x 2, 2x2 + 5x 3, and -a;2 4x + 4 generate P2(-R) since each of the three given polynomials belongs to P2(i?) and each polyno-mial ax2 + bx + c in P2(-R) is a linear combination of these three, namely, ( -8a + 56 + 3c)(x2 + 3x - 2) + (4a - 26 - c)(2a-2 + 5x - 3) + ( - a + 6 + c) ( -x 2 - 4a; + 4) = az2 + bx + a / Example 5 The matrices 1 1 1 0 1 1 0 1 1 0 1 1 and 0 1 1 1 generate M2x2(.ft) since an arbitrary matrix A in M2x2(.R) c a n be expressed as a linear combination of the four given matrices as follows: a n aV2 a2i a22 .1 1 1 2 . = (^ai l + o a 12+ oa21 ~ 3^22) ,1 3 a n 1 1 oa12 - ofl21 + oa22, ,1 2 1 1 , ,2 1 1 1 x + ('Q0!! + o a 12+ ofl21 + oa22) 1 0 1 ) 1 0 1 0 1 1 1 On the other hand, the matrices 1 0 0 1 1 1 0 1 and 1 0 1 1 32 Chap. 1 Vector Spaces do not generate M2x2(i?) because each of these matrices has equal diagonal entries. So any linear combination of these matrices has equal diagonal en-tries. Hence not every 2 x 2 matrix is a linear combination of these three matrices. At the beginning of this section we noted that the equation of a plane through three noncollinear points in space, one of which is the origin, is of the form x su + tv, where u, v R3 and s and t are scalars. Thus x G R3 is a linear combination of u, v E R if and only if x lies in the plane containing u and v. (See Figure 1.5.) Figure 1.5 Usually there are many different subsets that generate a subspace W. (See Exercise 13.) It is natural to seek a subset of W that generates W and is as small as possible. In the next section we explore the circumstances under which a vector can be removed from a generating set to obtain a smaller generating set. EXERCISES 1. Label the following statements as true or false. (a) The zero vector is a linear combination of any nonempty set of vectors. (b) The span of 0 is 0 . (c) If 5 is a subset of a vector space V, then span(5) equals the inter-section of all subspaces of V that contain S. (d) In solving a system of linear equations, it is permissible to multiply an equation by any constant. (e) In solving a system of linear equations, it is permissible to add any multiple of one equation to another. (f) Every system of linear equations has a solution. Sec. 1.4 Linear Combinations and Systems of Linear Equations 33 2. Solve the following systems of linear equations by the method intro-duced in this section. (a) (b) (c) (d) (e) (f) 2si - 1x2 - 3x3 = - 2 3xi 3x2 2x3 + 5x4 = 7 X X2 2x3 x4 = 3 3xi - lx2 + 4x3 = 10 x - 2x2 + X3 = 3 2x X2 2x3 = 6 x + 2x2 X3 + a*4 = 5 xi 4- 4x2 - 3x3 3x4 6 2xi 4- 3x2 X3 4- 4x4 = 8 xi 4- 2x2 + 2x3 = 2 xi + 8x3 + 5.T4 = - 6 xi 4- x2 4- 5.T3 + 5x4 = 3 Xi + 2x2 - 4x3 - X4 + x5 = 7 -Xi 4- l()x3 - 3x4 - 4x5 = -16 2xi + 5x2 - 5x3 ~ 4x4 - x5 = 2 4xi + l l x 2 - 7x3 - IOX4 - 2x5 = 7 Xi 4- 2x2 + 6x3 = - 1 2xi + x2 + x3 = 8 3xi + x2 - X3 = 15 xi 4- 3x2 + IOX3 = - 5 / 3. For each of the following lists of vectors in R3, determine whether the first vector can be expressed as a linear combination of the other two. (a) ( -2 ,0 ,3) , (1 ,3 ,0) , (2 ,4 , -1) (b) (1,2, -3) , ( -3 ,2 ,1) , ( 2 , - 1 , - 1 ) (c) (3,4,1), (1,-2,1) , ( - 2 , - 1 , 1 ) (d) (2,-1,0) , (1 ,2 , -3) , (1 , -3 ,2) (e) (5,1,-5) , ( 1 , - 2 , - 3 ) , ( - 2 , 3 , - 4 ) (f) ( -2,2,2) , (1 ,2 , -1) , ( - 3 , - 3 , 3 ) 4. For each list of polynomials in P3(K), determine whether the first poly-nomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f) x3 - 3x 4- 5, x3 4- 2x2 - x + 1, x3 + 3x2 - 1 4x3 + 2x2 - 6, x3 - 2x2 + 4x + 1,3x3 - 6x2 + x + 4 -2x3 - 1 lx2 4- 3x 4- 2, x3 - 2x2 + 3x - 1,2x3 + x2 + 3x - 2 x3 4- x2 + 2x 4-13,2x3 - 3x2 + 4x + 1, x3 - x2 + 2x 4- 3 x3 - 8x2 + 4x, x3 - 2x2 + 3x - 1, x3 - 2x + 3 6x3 - 3x2 4- x + 2, x3 - x2 + 2x 4- 3,2x3 - 3x 4- 1 34 Chap. 1 Vector Spaces 5. In each part, determine whether the given vector is in the span of S. (a) (2,-1,1), S = {(1,0,2), ( -1 ,1 ,1)} (b) (-1,2,1) , S = {(1,0,2), ( -1 ,1 ,1)} (c) ( -1,1,1,2) , S = {(1,0,1,-1), (0,1,1,1)} (d) ( 2 , - 1 , 1 , - 3 ) , S = {(1,0,1,-1), (0,1,1,1)} (e) - x 3 4-2x2 4-3x 4-3, S = {x3 4- x2 4-x + l , x 2 4- x 4- 1,X 4- 1} x + 3, S= {x3 + x2 + x + l ,x 2 4-x 4- l ,x4- 1} (f) 2 x 3 - x 2 (g) 1 2 - 3 4 (h) 1 0 0 1 s = S = 1 0 - 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 6. Show that the vectors (1,1,0), (1,0,1), and (0,1,1) generate F3. 7. In Fn, let ej denote the vector whose j t h coordinate is 1 and whose other coordinates are 0. Prove that {ei, e 2 , . . . , en} generates Fn. 8. Show that Pn(F) is generated by {1, x , . . . , xn}. 9. Show that the matrices 1 0 0 0 generate M2X2(F)-10. Show that if 1 0 0 1 0 0 0 0 1 0 and 0 0 0 1 Mi = 0 0 M2 = 0 0 0 1 and M3 = 0 1 1 0 then the span of {Mi, M2, M3} is the set of all symmetric 2 x 2 matrices. 11.T Prove that span({x}) = {ax: a F} for any vector x in a vector space. Interpret this result geometrically in R3. 12. Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W. 13. ^ Show that if Si and S2 are subsets of a vector space V such that Si C 52 , then span(Si) C span(52). In particular, if Si C 52 and span(Si) = V, deduce that span(52) V. 14. Show that if Si and S2 are arbitrary subsets of a vector space V, then span(S'iU5,2) = span(5i)+span(S'2). (The sum of two subsets is defined in the exercises of Section 1.3.) Sec. 1.5 Linear Dependence and Linear Independence 35 15. Let S and S2 be subsets of a vector space V. Prove that span (Si DS2) C span(Si) n span(S2). Give an example in which span(Si D S2) and span(Si) nspan(S2) are equal and one in which they are unequal. 16. Let V be a vector space and S a subset of V with the property that whenever v, 1)2,... ,vn S and aiVi 4- a2v2 4- 4- anvn = 0, then o = 0,2 = = an = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S. 17. Let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S gen-erates W? 1.5 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE Suppose that V is a vector space over an infinite field and that W is a subspace of V. Unless W is the zero subspace, W is an infinite set. It is desirable to find a 'small' finite subset S that generates W because we can then describe each vector in W as a linear combination of the finite number of vectors in S. Indeed, the smaller that S is, the fewer computations that are required to represent vectors in W. Consider, for example, the subspace W of R3 generated by S = {u,U2,^3,^/4}, where ui (2,-1,4) , u2 = (1,-1,3) , U3 = (1,1, 1), and U4 = (1, 2, 1). Let us attempt to find a proper subset of S that also generates W. The search for this subset is related to the question of whether or not some vector in S is a linear combination of the other vectors in S. Now U4 is a linear combination of the other vectors in S if and only if there arc scalars a, a2 , and a3 such that u4 = aiui 4- a2w2 4- a3w3, that is, if and only if there are scalars a i ,a 2 , and a3 satisfying (1, - 2 , -1 ) = (2ai 4- a2 4- a3 , ai - a2 4- a3 ,4a 1 4- 3a2 - a3). Thus U4 is a linear combination of ui ,u2 , and U3 if and only if the system of linear equations 2ai 4- a2 4- a3 = 1 - a i - a2 4- a3 = - 2 4ai 4- 3a2 03 = 1 has a solution. The reader should verify that no such solution exists. This does not, however, answer our question of whether some vector in S is a linear combination of the other vectors in S. It can be shown, in fact, that U3 is a linear combination of wi,u2, and U4, namely, u 3 2u ~ 3w2 4- OW4. 36 Chap. 1 Vector Spaces In the preceding example, checking that some vector in S is a linear combination of the other vectors in S could require that we solve several different systems of linear equations before we determine which, if any, of ui,U2,U3, and U4 is a linear combination of the others. By formulating our question differently, we can save ourselves some work. Note that since U3 = 2ui 3w2 4- OW4, we have 2ui 4- 3u2 4- w3 OW4 = 0. That is, because some vector in S is a linear combination of the others, the zero vector can be expressed as a linear combination of the vectors in S using coefficients that are not all zero. The converse of this statement is also true: If the zero vector can be written as a linear combination of the vectors in S in which not all the coefficients are zero, then some vector in S is a linear combination of the others. For instance, in the example above, the equation 2wi 4- 3u2 4- 3 OW4 = 0 can be solved for any vector having a nonzero coefficient; so U, w2, or U3 (but not U4) can be written as a linear combination of the other three vectors. Thus, rather than asking whether some vector in S is a linear combination of the other vectors in S, it is more efficient to ask whether the zero vector can be expressed as a linear combination of the vectors in S with coefficients that are not all zero. This observation leads us to the following definition. Definition. A subset S of a vector space V is called linearly dependent if there exist a finite number of distinct vectors W], u 2 , . . . ,un in S and scalars ai, a 2 , . . . , a n , not all zero, such that aiUi 4- a2w2 4- + anur = 0. In this case we also say that the vectors of S are linearly dependent. For any vectors Wi, w 2 , . . . , un, we have aiUi 4- a2w2 4- 4- anun = 0 if ai = a-2 = = a n = 0. We call this the trivial representation of 0 as a linear combination of u, w 2 , . . . , un. Thus, for a set to be linearly depen-dent, there must exist a nontrivial representation of 0 as a linear combination of vectors in the set. Consequently, any subset of a vector space that con-tains the zero vector is linearly dependent, because 0 = 1 0 is a nontrivial representation of 0 as a linear combination of vectors in the set. Example 1 Consider the set S = {(1,3, -4 ,2) , (2,2, -4 ,0) , (1, -3 ,2 , -4) , ( -1,0,1,0)} in R4. We show that S is linearly dependent and then express one of the vectors in S as a linear combination of the other vectors in S- To show that Sec. 1.5 Linear Dependence and Linear Independence 37 S is linearly dependent, we must find scalars ai, a2 , a3, and 04, not all zero, such that a i ( l , 3 , - 4 , 2 ) 4 - a 2 ( 2 , 2 , - 4 , 0 ) 4 - a 3 ( l , - 3 , 2 , - 4 ) 4 - a 4 ( - l , 0 , l , 0 ) = 0. Finding such scalars amounts to finding a nonzero solution to the system of linear equations ai 4- 2a2 4- a 3 a4 = 0 3ai 4- 2a2 3a3 = 0 4ai 4a2 4- 2a3 4- a4 = 0 2ai 4a3 = 0. One such solution is ai = 4, a2 = 3, 03 = 2, and 04 = 0. Thus S is a linearly dependent subset of R4, and 4(1,3, -4 ,2 ) - 3(2,2, -4 ,0 ) + 2(1, -3 ,2 , - 4 ) 4- 0 ( - l , 0,1,0) = 0. Example 2 In M2x3(i?), the set 1 - 3 2 -4 0 5 is linearly dependent because - 3 7 4 6 - 2 - 7 - 2 3 - 1 - 3 11 2 1 - 3 2 - 4 0 5 4-3 -3 7 4 _ (-2 3 11 6 - 2 -7 V1 3 2 0 0 0 0 0 0 Definition. A subset S of a vector space that is not linearly dependent is called linearly independent. As before, we also say that the vectors of S are linearly independent. The following facts about linearly independent sets are true in any vector space. 1. The empty set is linearly independent, for linearly dependent sets must be nonempty. 2. A set consisting of a single nonzero vector is linearly independent. For if {u} is linearly dependent, then au = 0 for some nonzero scalar a. Thus u = a (au) a 0 = 0. 3. A set is linearly independent if and only if the only representations of 0 as linear combinations of its vectors are trivial representations. 38 Chap. 1 Vector Spaces The condition in item 3 provides a useful method for determining whether a finite set is linearly independent. This technique is illustrated in the exam-ples that follow. Example 3 To prove that the set S = {(1,0,0,-1), (0 ,1,0,-1) , (0 ,0,1,-1) , (0,0,0,1)} is linearly independent, we must show that the only linear combination of vectors in S that equals the zero vector is the one in which all the coefficients are zero. Suppose that ai, 0,2,0,3, and 04 are scalars such that a i ( l , 0 , 0 , - 1 ) 4 - a 2 ( 0 , l , 0 , - l ) + a 3 ( 0 , 0 , 1 , - 1 ) + a4(0,0,0,1) = (0,0,0,0). Equating the corresponding coordinates of the vectors on the left and the right sides of this equation, we obtain the following system of linear equations. ai = 0 a2 = 0 a 3 = 0 ai a2 03 4- 04 = 0 Clearly the only solution to this system is ai = a2 = a 3 = a4 = 0, and so S is linearly independent. Example 4 For k = 0 , 1 , . . . , n let pk(x) = xk + xk+1 4- 4- xn . The set {po(x),pi(x),...,pn(x)} is linearly independent in Pn(F). For if a0p0(x) 4- aipi(x) H 1- anpn(x) = 0 for some scalars ao, a i , . . . , a n , then ao 4- (a0 4- ai)x 4- (a0 4- ai 4- a2)x2 -| 1- (a0 4- ai -I h a n )x n = 0. By equating the coefficients of xk on both sides of this equation for k = 1,2 , . . . , n, we obtain ao a0 4- ai ao 4- a 4- a2 ao 4- ai 4- a2 4- = 0 = 0 = 0 4- a n = 0 Clearly the only solution to this system of linear equations is ao = ai = = a n = 0. Sec. 1.5 Linear Dependence and Linear Independence 39 The following important results are immediate consequences of the defi-nitions of linear dependence and linear independence. Theorem 1.6. Let V be a vector space, and let S] C S2 C V. If Si is linearly dependent, then S2 is linearly dependent. Proof. Exercise. SSI Corollary. Let V be a vector space, and let St C S2 C V. If S2 is linearly independent, then Si is linearly independent. Proof. Exercise. I Earlier in this section, we remarked that the issue of whether S is the smallest generating set for its span is related to the question of whether some vector in S is a linear combination of the other vectors in S. Thus the issue of whether S is the smallest generating set for its span is related to the question of whether S is linearly dependent. To see why. consider the subset S = {u,1*2,^3,114} of R3, where //, = (2 , -1 ,4) , w2 = (1, 1,3), 113 = (1,1,-1) , and 4 ( 1 , - 2 , - 1 ) . We have previously noted that S is linearly dependent: in fact. -2u. 4- 3u2 4- U3 - O//4 = 0. This equation implies that w3 (or alternatively, U or u2) is a linear combina-tion of the other vectors in S. For example, 11.3 = 2ii 3u2 4- OU4. Therefore every linear combination aUi f a2u2 + a3':i + 04114 of vectors in S can be written as a linear combination of U[, u2, and 114: atii 4- a.2u2 4- a3?i3 4- 0.4U4 = aiUi + o.2u2 4- a3(2ui 3u2 4- OM4) 4- 0411.4 = (ai 4- 2a3)jvi + (a2 - 3a3)2 + 04^4. Thus the subset S' {ai,u2,U4} of S has the same span as S! More generally, suppose that S is any linearly dependent set containing two or more vectors. Then some vector v S can be written as a linear combination of the other vectors in S. and the subset obtained by removing v from S has the same span as S. It follows that if no proper subset of S generates the span of S, then S must be linearly independent. Another way to view the preceding statement is given in Theorem 1.7. Theorem 1.7. Let S be a linearly independent subset of a vector space V, and let V be a vector in V that is not in S. Then S U {v} is linearly dependent if and only ifv span(S). 40 Chap. 1 Vector Spaces Proof. If Su{v} is linearly dependent, then there are vectors u, u2,..., un in S U {v} such that a i^i 4- a2u2 4- + anun = 0 for some nonzero scalars a i , a 2 , . . . , a n . Because S is linearly independent, one of the u^s, say i, equals v. Thus aiv 4- a2u2 4- 4- anun = 0, and so v = a1 (-a2u2 onun) = - ( a x a2)u2 (ax 1an)un. Since v is a linear combination of u2,... ,un, which are in S, we have v span(S). Conversely, let v span(S). Then there exist vectors Vi,V2, ,vm in S and scalars 6i, 62, . . . , 6m such that v = biVi + b2v2 4- 4- bmvm. Hence 0 = bvi + b2v2 4- -l)v. Since v ^ Vi for i = 1 ,2 , . . . , ra, the coefficient of v in this linear combination is nonzero, and so the set {vi,v2,... ,vm,v} is linearly dependent. Therefore S U {v} is linearly dependent by Theorem 1.6. H Linearly independent generating sets are investigated in detail in Sec-tion 1.6. EXERCISES 1. Label the following statements as true or false. (a) If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S. (b) Any set containing the zero vector is linearly dependent. (c) The empty set is linearly dependent. (d) Subsets of linearly dependent sets are linearly dependent. (e) Subsets of linearly independent sets are linearly independent. (f) If aixi 4- a2x2 4- 4- anxn = 0 and Xi ,x 2 , . . . ,xn are linearly independent, then all the scalars a are zero. 2. Determine whether the following sets are linearly dependent or linearly independent. 1 - 3 - 2 4 (a) - 2 6 4 - 8 in M2 x 2( f l ) ^ H - l A ) 2 _ 4 J ) - M 2 >< 2 ^ (c) {x3 4- 2x2, - x 2 4- 3.x 4- 1, x3 - x2 4- 2x - 1} in P3(R) 3The computations in Exercise 2(g), (h), (i), and (j) are tedious unless technology is used. Sec. 1.5 Linear Dependence and Linear Independence 41 (d) {x3 - x, 2x2 4- 4, - 2 x 3 4- 3x2 4- 2x 4- 6} in P3(R) (e) { ( l , - l , 2 ) , ( l , - 2 , l ) , ( l , l , 4 ) } i n R 3 (f) { ( l , - l , 2 ) , ( 2 , 0 , l ) , ( - l , 2 , - l ) } i n R 3 (g) (h) 1 0 ~2 1 1 0 ~2 1 0 - 1 1 1 0 - 1 1 1 - 1 2 1 0 - 1 2 1 0 2 1 - 4 4 2 1 2 - 2 in M2x2(tf) in M 2 x 2 (#) (i) {x4 - x3 4- 5x2 - 8x 4- 6, - x 4 4-x3 - 5x2 + 5x - 3, x4 4- 3x2 - 3x 4- 5,2x4 4- 3x3 4- 4x2 - x 4-1, x3 - x 4- 2} in P4(R) (j) {x4 - x3 + ox2 - 8x 4- 6, - x 4 4- x3 - 5x2 4- 5x - 3, x4 + 3x2 - 3x 4- 5,2x4 4- x3 4- 4x2 4- 8x} in P4(R) 3. In M3 x 2(F), prove that the set is linearly dependent. / 4. In Fn, let ej denote the vector whose j th coordinate is 1 and whose other coordinates are 0. Prove that {e i ,e 2 , . . . , e n } is linearly independent. 5. Show that the set { l , x , x 2 , . . . , x n } is linearly independent in P n (F) . 6. In M m X n (F) , let E1* denote the matrix whose only nonzero entry is 1 in the zth row and j th column. Prove that Exi: 1 < i < m, 1 < j < n) is linearly independent. 7. Recall from Example 3 in Section 1.3 that the set of diagonal matrices in M2x2(^) is a subspace. Find a linearly independent set that generates this subspace. 8. Let S = {(1,1,0), (1,0,1), (0,1,1)} be a subset of the vector space F3. (a) Prove that if F = R, then S is linearly independent. (b) Prove that if F has characteristic 2, then S is linearly dependent. 9.* Let u and v be distinct vectors in a vector space V. Show that {u, v} is linearly dependent if and only if u or v is a multiple of the other. 10. Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another. 42 Chap. 1 Vector Spaces 11. Let S = {ui,U2, ,Wn} be a linearly independent subset of a vector space V over the field Z2. How many vectors are there in span(S)? Justify your answer. 12. Prove Theorem 1.6 and its corollary. 13. Let V be a vector space over a field of characteristic not equal to two. (a) Let u and v be distinct vectors in V. Prove that {u, v} is linearly independent if and only if {u 4- v, u v} is linearly independent. (b) Let u, v, and w be distinct vectors in V. Prove that {u,v,w} is linearly independent if and only if {u + v,u + w, v 4- w} is linearly independent. 14. Prove that a set S is linearly dependent if and only if S = {0} or there exist distinct vectors v,U,U2, , un in S such that v is a linear combination of ui,u2,...,un. 15. Let S = {ui,u2,... ,un} be a finite set of vectors. Prove that S is linearly dependent if and only if ui = 0 or Wfc+i span({tii, w 2 , . . . , Uk}) for some k (1 < k < n). 16. Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. 17. Let M be a square upper triangular matrix (as defined in Exercise 12 of Section 1.3) with nonzero diagonal entries. Prove that the columns of M are linearly independent. 18. Let S be a set of nonzero polynomials in P(F) such that no two have the same degree. Prove that S is linearly independent. 19. Prove that if {Ai,A2, ,Ak} is a linearly independent subset of Mnxn(-F)j then {A, A2, - , Alk} is also linearly independent. 20. Let / , g, F(R, R) be the functions defined by f(t) = ert and g(t) = est, where r ^ s. Prove that / and g are linearly independent in J-(R, R). 1.6 BASES AND DIMENSION We saw in Section 1.5 that if S is a generating set for a subspace W and no proper subset of S is a generating set for W, then S must be linearly independent. A linearly independent generating set for W possesses a very useful propertyevery vector in W can be expressed in one and only one way as a linear combination of the vectors in the set. (This property is proved below in Theorem 1.8.) It is this property that makes linearly independent generating sets the building blocks of vector spaces. Sec. 1.6 Bases and Dimension 43 Definition. A basis j3 for a vector space V is a linearly independent subset of V that generates V. If ft is a basis for V, we aiso say that the vectors of ft form a basis for V. Example 1 Recalling that span(0) = {()} and 0 is linearly independent, we see that 0 is a basis for the zero vector space. Example 2 In Fn, let ei = (1,0,0, . . . ,0),e2 = (0,1,0,. . . , 0 ) , . . . ,e = (0,0, . . . ,0,1); {ei, e 2 , . . . , e n } is readily seen to be a basis for Fn and is called the standard basis for Fn. Example 3 In MTOXn(F), let E*i denote the matrix whose only nonzero entry is a 1 in the ith. row and j th column. Then {E*? : 1 < i < m, 1 < j < n} is a basis for Mm x n(F) . ' Example 4 In Pn(F) the set {1, x, x 2 , . . . , x n } is a basis. We call this basis the s t anda rd basis for Pn(F). / Example 5 In P(F) the set {1 ,x ,x 2 , . . .} is a basis. Observe that Example 5 shows that a basis need not be finite. In fact, later in this section it is shown that no basis for P(F) can be finite. Hence not every vector space has a finite basis. The next theorem, which is used frequently in Chapter 2, establishes the most significant property of a basis. Theorem 1.8. Let V be a vector space and ft = {ui,u2,... ,un} be a subset ofV. Then ft is a basis for V if and only if each o G V can be uniquely expressed as a linear combination of vectors of ft, that is, can be expressed in the form v = aiUi 4- a2u2 H - anun for unique scalars a, a 2 , . . . , a n . Proof. Let ft be a basis for V. If v e V, then v span(/?) because span(/i/) = V. Tims v is a linear combination of the vectors of ft. Suppose that aU +a2u2 4- anun and v = bU + b2u2 + + bnut 44 Chap. 1 Vector Spaces are two such representations of v. Subtracting the second equation from the first gives 0 = (a - b)ui + (a2 - b2)u2 4- h (an - bn)un. Since ft is linearly independent, it follows that ai b = a2 b2 = = on - bn = 0. Hence a l>i,a2 = b2,---,an = bn, and so v is uniquely expressible as a linear combination of the vectors of ft. The proof of the converse is an exercise. I Theorem 1.8 shows that if the vectors u,u2,... ,un form a basis for a vector space V, then every vector in V can be uniquely expressed in the form v = arui 4- a2u2 - h anun for appropriately chosen scalars 0],a2,... ,an. Thus v determines a unique n-tuple of scalars ( a i , a 2 , . . . , an) and, conversely, each n-tuple of scalars de-termines a unique vector v V by using the entries of the n-tuple as the coefficients of a linear combination of ui,u2,..., un. This fact suggests that V is like the vector space Fn, where n is the number of vectors in the basis for V. We see in Section 2.4 that this is indeed the case. In this book, we are primarily interested in vector spaces having finite bases. Theorem 1.9 identifies a large class of vector spaces of this type. Theorem 1.9. If a vector space V is generated by a Bnite set S, then some subset of S is a basis for V. Hence V has a finite basis. Proof. If S = 0 or S = {0}, then V = {()} and 0 is a subset of S that is a basis for V. Otherwise S contains a nonzero vector u. By item 2 on page 37, {ui} is a linearly independent set. Continue, if possible, choosing vectors u2,...,itfc in S such that {ui,u2,.... Uk} is linearly independent. Since S is a finite set, we must eventually reach a stage at which ft = {ui,U2,..., Uk) is a linearly independent subset of S, but adjoining to ft any vector in S not in ft produces a linearly dependent set. We claim that ft is a basis for V, Because ft is linearly independent by construction, it suffices to show that ft spans V. By Theorem 1.5 (p. 30) we need to show that S C span(/?). Let v S. If v ft, then clearly v span(/3). Otherwise, if v ft, then the preceding construction shows that ft U {v} is linearly dependent. So v span(/3) by Theorem 1.7 (p. 39). Thus S C span(/?). I Because of the method by which the basis ft was obtained in the proof of Theorem 1.9, this theorem is often remembered as saying that a tinite spanning set for V can be reduced to a basis for V. This method is illustrated in the next example. Sec. 1.6 Bases and Dimension 45 Example 6 Let S = {(2, -3 .5) , (8. -12.20), (1,0, -2) . (0.2, -1) , (7.2.0)}. It can be shown that S generates R3. We can select a basis for R3 that is a subset of S by the technique used in proving Theorem 1.9. To start, select any nonzero vector in S, say (2. - 3 . 5), to be a vector in the basis. Since 4(2,-3,5) = (8,-12.20), the set {(2,-3,5), (8,-12,20)} is linearly dependent by Exercise 9 of Section 1.5. Hence we do not include (8, 12,20) in our basis. On the other hand, (1,0, - 2 ) is not a multiple of (2, -3 ,5 ) and vice versa, so that the set {(2. 3. 5), (1,0. 2)} is linearly independent. Thus we include (1,0, 2) as part of our basis. Now we consider the set {(2. -3 .5) , (1.0, -2 ) . (0 ,2 , -1)} obtained by ad-joining another vector in S to the two vectors that we have already included in our basis. As before, wc include (0,2, 1) in our basis or exclude it from the basis according to whether {(2, -3 ,5) . (1.0, -2) , (0,2, -1 )} is linearly in-dependent or linearly dependent. An easy calculation shows that this set is linearly independent, and so we include (0,2, 1) in our basis. In a similar fashion the final vector in S is included or excluded from our basis according to whether the set {(2,-3,5), (1 ,0 , -2) , (0 ,2 , -1) , (7,2,0)} is linearly independent or linearly dependent. Because 2(2, -3 ,5 ) 4- 3(1,0, - 2 ) 4- 4(0. 2. -1 ) - (7. 2,0) = (0,0.0), we exclude (7,2,0) from our basis. We conclude that { (2 . -3 .5 ) , (1 .0 , -2 ) . (0 ,2 , -1 )} is a subset of S that is a basis for R3. The corollaries of the following theorem are perhaps the most significant results in Chapter 1. Theorem 1.10 (Replacement Theorem). Let V be a vector space that is generated by a set G containing exactly n vectors, and let L be a linearly independent subset of V containing exactly m vectors. Then rn < n and there exists a subset HofG containing exactly n rn vectors such that L U H generates V. Proof. The proof is by mathematical induction on m. The induction begins with m = 0; for in this case L = 0 , and so taking H = G gives the desired result. 46 Chap. 1 Vector Spaces Now suppose that the theorem is true for some integer m > 0. We prove that the theorem is true for m 4- 1. Let L = {v,V2, ,v m + i } be a linearly independent subset of V consisting of ra 4- 1 vectors. By the corollary to Theorem 1.6 (p. 39), {v,v2,... ,vm} is linearly independent, and so we may apply the induction hypothesis to conclude that rn < n and that there is a subset {ui ,u2,..., un-m } of G such that {vi ,v2,... ,vm}u{ui,u2,..., un-m } generates V. Thus there exist scalars a, a 2 , . . . , aTO, b, b2,..., bn-m such that aV 4- a2v2 + amvrn 4- biui 4- b2u2 4- bn-nrn **nrn = V TO+1 (9) Note that n m > 0, lest vm+i be a linear combination of v, v%,..., vm, which by Theorem 1.7 (p. 39) contradicts the assumption that L is linearly independent. Hence n > ra; that is, n > rn + l. Moreover, some bi, say &i, is nonzero, for otherwise we obtain the same contradiction. Solving (9) for wi gives ui = (-6^1Oi)1 4- (-bi1a2)v2 + 4- (-bilam)vm + (lhl)vni+i 4- (-bi1b2)u2 H 4- (-6r/16n_m)u f l_m . Let H = {u2,..., Un-m). Then ui span(LUi/), and because V, v2,..., um , u 2 , . . . , un-m are clearly in span(L U H), it follows that {vi, v2,..., vm,ui,u2,..., Un-m} Q span(L U H). Because {v,v2,..., vm, ui, u2,..., w n - m } generates V, Theorem 1.5 (p. 30) implies that span(L U H) = V. Since H is a subset of G that contains (n rn) 1 = n (m + 1) vectors, the theorem is true for rn 4- 1. This completes the induction. 1 Corollary 1. Let V be a vector space having a tinitc basis. Then every basis for V contains the same number of vectors. Proof. Suppose that ft is a finite basis for V that contains exactly n vectors, and let 7 be any other basis for V. If 7 contains more than n vectors, then we can select a subset S of 7 containing exactly n 4-1 vectors. Since S is linearly independent and ft generates V, the replacement theorem implies that n +1 < n, a contradiction. Therefore 7 is finite, and the number m of vectors in 7 satisfies 77?. < n. Reversing the roles of ft and 7 and arguing as above, we obtain n < ra. Hence rn = n. I If a vector space has a finite basis, Corollary 1 asserts that the number of vectors in any basis for V is an intrinsic property of V. This fact makes possible the following important definitions. Definitions. A vector space is called finite-dimensional if it has a basis consisting of a Unite number of vectors. The unique number of vectors Sec. 1.6 Bases and Dimension 47 in each basis for V is called the dimension of/ and is denoted by dim(V). A vector space that is not finite-dimensional is called infinite-dimensional. The following results are consequences of Examples 1 through 4. Example 7 The vector space {0} has dimension zero. Example 8 The vector space Fn has dimension n. Example 9 The vector space M m x n ( F ) has dimension rnn. Example 10 The vector space Pn{F) has dimension n + 1. The following examples show that the dimension of a vector space depends on its field of scalars. / Example 11 Over the field of complex numbers, the vector space of complex numbers has dimension 1. (A basis is {1}.) Example 12 Over the field of real numbers, the vector space of complex numbers has dimension 2. (A basis is {l,i}.) In the terminology of dimension, the first conclusion in the replacement theorem states that if V is a finite-dimensional vector space, then no linearly independent subset of V can contain more than dim(V) vectors. From this fact it follows that the vector space P(F) is infinite-dimensional because it has an infinite linearly independent set, namely {l,x, x2,...}. This set is, in fact, a basis for P(F). Yet nothing that we have proved in this section guarantees an infinite-dimensional vector space must have a basis. In Section 1.7 it is shown, however, that every vector space has a basis. Just as no linearly independent subset of a finite-dimensional vector space V can contain more than dim(V) vectors, a corresponding statement can be made about the size of a generating set. Corollary 2. Let V be a vector space with dimension n. (a) Any finite generating set for V contains at least n vectors, and a gener-ating set for V that contains exactly n vectors is a basis for V. 48 Chap. 1 Vector Spaces (b) Any linearly independent subset ofV that contains exactly n vectors is a basis for V. (c) Every linearly independent subset of V can be extended to a basis for V. Proof. Let ft be a basis for V. (a) Let G be a finite generating set for V. By Theorem 1.9 some subset H of G is a basis for V. Corollary 1 implies that H contains exactly n vectors. Since a subset of G contains n vectors, G must contain at least n vectors. Moreover, if G contains exactly n vectors, then we must have H = G, so that G is a basis for V. (b) Let L be a linearly independent subset of V containing exactly n vectors. It follows from the replacement theorem that there is a subset H of ft containing n n = 0 vectors such that L U H generates V. Thus H = 0 , and L generates V. Since L is also linearly independent, L is a basis for V. (c) If L is a linearly independent subset of V containing ra vectors, then the replacement theorem asserts that there is a subset H of ft containing exactly n ra vectors such that LU H generates V. Now L U H contains at most n vectors; therefore (a) implies that L U H contains exactly n vectors and that L U H is a basis for V. 1 Example 13 It follows from Example 4 of Section 1.4 and (a) of Corollary 2 that {x2 4- 3x - 2,2x2 4- 5x - 3, - x 2 - 4x 4- 4} is a basis for P2(R.). Example 14 It follows from Example 5 of Section 1.4 and (a) of Corollary 2 that : ; ) ( , : : ) ( : : ) . ( : : is a basis for M2x2(R). ^ Example 15 It follows from Example 3 of Section 1.5 and (b) of Corollary 2 that {(1,0,0, -1) , (0,1,0, -1) , (0,0,1, -1) , (0,0,0,1)} is a basis for R4. Sec. 1.6 Bases and Dimension Example 16 49 For k = 0 ,1 , . . . , n, let pk(x) = xfc4-xfc+14 of Section 1.5 and (b) of Corollary 2 that f xn. It follows from Example 4 {po(x),Pi(x),...,pn(x)} is a basis for Pn(F) . A procedure for reducing a generating set to a basis was illustrated in Example 6. In Section 3.4, when we have learned more about solving systems of linear equations, we will discover a simpler method for reducing a gener-ating set to a basis. This procedure also can be used to extend a linearly independent set to a basis, as (c) of Corollary 2 asserts is possible. An Overview of Dimension and Its Consequences Theorem 1.9 as well as the replacement theorem and its corollaries contain a wealth of information about the relationships among linearly independent sets, bases, and generating sets. For this reason, we summarize here the main results of this section in order to put them into better/perspective. A basis for a vector space V is a linearly independent subset of V that generates V. If V has a finite basis, then every basis for V contains the same number of vectors. This number is called the dimension of V, and V is said to be finite-dimensional. Thus if the dimension of V is n, every basis for V contains exactly n vectors. Moreover, every linearly independent subset of V contains no more than n vectors and can be extended to a basis for V by including appropriately chosen vectors. Also, each generating set for V contains at least n vectors and can be reduced to a basis for V by excluding appropriately chosen vectors. The Venn diagram in Figure 1.6 depicts these relationships. Figure 1.6 50 Chap. 1 Vector Spaces The Dimension of Subspaces Our next result relates the dimension of a subspace to the dimension of the vector space that contains it. Theorem 1.11. Let W be a subspace of a finite-dimensional vector space V. Then W is finite-dimensional and dim(W) < dim(V). Moreover, if dim(W) = dim(V), then V = W. Proof. Let dim(V) = n. If W = {()}, then W is finite-dimensional and dim(W) = 0 < n. Otherwise, W contains a nonzero vector Xj; so {xi} is a linearly independent set. Continue choosing vectors, Xi,X2, , X& in W such that {xi,X2, ,Xfc} is linearly independent. Since no linearly independent subset of V can contain more than n vectors, this process must stop at a stage where A; < n and {x i ,x 2 , . . . ,Xfc} is linearly independent but adjoining any other vector from W produces a linearly dependent set. Theorem 1.7 (p. 39) implies that {x i ,x 2 , . . . ,Xfc} generates W, and hence it is a basis for W. Therefore dim(W) = k < n. If dim(W) = n, then a basis for W is a linearly independent subset of V containing n vectors. But Corollary 2 of the replacement theorem implies that this basis for W is also a basis for V; so W = V. 1 Example 17 Let / W = {(o.,a2,a:i,04,a&) F5: a. + 03 4- 05 = 0, a2 = 04). It is easily shown that W is a subspace of F5 having {(-1,0,1,0,0), (-1,0,0,0,1) , (0,1,0,1,0)} as a basis. Thus dim(W) = 3. Example 18 The set of diagonal n x n matrices is a subspace W of M n x n ( F ) (see Example 3 of Section 1.3). A basis for W is {En,E22,...,Enn}, where E%3 is the matrix in which the only nonzero entry is a 1 in the iih row and j th column. Thus dim(W) = n. Example 19 We saw in Section 1.3 that the set of symmetric n x n matrices is a subspace W of M n X n (F) . A basis for W is {Aij : 1 < i < j < n}, Sec. 1.6 Bases and Dimension 51 where A*3 is the n x n matrix having 1 in the ith row and jfth column, 1 in the j th row and ith column, and 0 elsewhere. It follows that dim(W) =n + (n~ 1)4- 4- 1 = -n(n+ 1 Corollary. If W is a subspace of a finite-dimensional vector space V. then any basis for W can be extended to a basis for V. Proof. Let S be a basis for W. Because S is a linearly independent, subset of V, Corollary 2 of the replacement theorem guarantees that S can be extended to a basis for V. J Example 20 The set of all polynomials of the form a 1 8 xl 8 4-a 1 6 x 1 6 4- 4- a2x 4- a0 . where ais,ai6, ,a2 ,an F , is a subspace W of Pis(F). A basis for W is {1. x 2 , . . . , x16 , x18}, which is a subset of the standard basis for Pig(F). We can apply Theorem 1.11 to determine the subspaces of R2 and R Since R2 has dimension 2, subspaces of R2 can be of dimensions 0, 1, or 2 only. The only subspaces of dimension 0 or 2 are {0} and R2. respectively. Any subspace of R2 having dimension 1 consists of all scalar multiples of some nonzero vector in R2 (Exercise 11 of Section 1.4). If a point of R2 is identified in the natural way with a point in the Euclidean plane, then it is possible to describe the subspaces of R geometrically: A subspace of R2 having dimension 0 consists of the origin of the Euclidean plane, a subspace of R with dimension 1 consists of a line through the origin, and a subspace of R2 having dimension 2 is the entire Euclidean plane. Similarly, the subspaces of R,} must have dimensions 0, 1, 2, or 3. Inter-preting these possibilities geometrically, we see that a subspace of dimension zero must be the origin of Euclidean 3-space, a subspace of dimension 1 is a line through the origin, a subspace of dimension 2 is a plane through the origin, and a subspace of dimension 3 is Euclidean 3-space itself. The Lagrange Interpolation Formula Corollary 2 of the replacement theorem can be applied to obtain a useful formula. Let Cq,Ci,...,cn be distinct scalars in an infinite field F. The polynomials fo(x). fi(x) fn(x) defined by fi(*) = (X - Cp) - (X - Cj_!)(x - Cj+i) (X - Cn) {(< -Co)--- (Ci - Ci~l)(Ci - Ci+l) (C-i- Cn) n fc=0 kjti X -Cfc Ci - Cfc 52 Chap. 1 Vector Spaces are called the Lagrange polynomials (associated with co,ci , . . . ,cn). Note that each fi(x) is a polynomial of degree n and hence is in Pn(F) . By re-garding fi(x) as a polynomial function /*: F > F, we see that fi(cj) = 0 tii^j 1 if i = j. (10) This property of Lagrange polynomials can be used to show that ft = {/o> /i) > fn} is a linearly independent subset of Pn(F) . Suppose that /_, aifi 0 fr s o m e scalars an, a i , . . . , an , i=0 where 0 denotes the zero function. Then n ^2 aifi(cj) = 0 for j = 0 , 1 , . . . , n. i=0 But also n 22a.ifi(cj) = Oj i=0 by (10). Hence aj = 0 for j = 0 ,1 , . . . , n; so /3 is linearly independent. Since the dimension of P n (F) is n+1, it follows from Corollary 2 of the replacement theorem that ft is a basis for P n (F) . Because ft is a basis for P n(F) , every polynomial function g in P n (F) is a linear combination of polynomial functions of ft, say, n 9 = ^>2bifi. 1=0 It follows that 9(cj) = ^2bifi(cj) = bJ'^ i=0 so 9 = ^9(ci)h i-0 is the unique representation of g as a linear combination of elements of ft. This representation is called the Lagrange interpolation formula. Notice Sec. 1.6 Bases and Dimension 53 that the preceding argument shows that if bo,b,...,bn are any n4-1 scalars in F (not necessarily distinct), then the polynomial function n 9 = Y1 Mi i=0 is the unique polynomial in P n (F) such that g(cj) = bj. Thus we have found the unique polynomial of degree not exceeding n that has specified values bj at given points Cj in its domain (j = 0 ,1 , . . . ,n) . For example, let us construct the real polynomial g of degree at most 2 whose graph contains the points (1,8), (2,5), and (3, 4). (Thus, in the notation above, cn = 1, C = 2, C2 = 3, bo = 8, &i = 5, and b2 = 4.) The Lagrange polynomials associated with Co, Ci, and c2 are , . , (x - 2)(x - 3) l , o . /^) = ( l - 2 ) ( l - 3 ) - 2 - ^ - 5 - + 6 ^ / ' (') = t l i t ^ = -Hx2 - 4x + 3), ( 2 - l ) ( 2 - 3 ) and / / * < * ) = ( r ? i ! r ? M ( * 2 - 3 * + 2 ) . (3 - 1)(3 - 2) 2 Hence the desired polynomial is 2 g(x) = J2bifi(x) = 8/0(x) 4- 5/i(x) - 4/2(x) i=0 = 4(x2 - 5x 4- 6) - 5(x2 - 4x 4- 3) - 2(x2 - 3x 4- 2) = -3x 2 + 6x 4- 5. An important consequence of the Lagrange interpolation formula is the fol-lowing result: If / P n (F) and f(ci) = 0 for n4T distinct scalars CQ, C, ..., cn in F, then / is the zero function. EXERCISES 1. Label the following statements as true or false. (a) The zero vector space has no basis. (b) Every vector space that is generated by a finite set has a basis. (c) Every vector space has a finite basis. (d) A vector space cannot have more than one basis. 54 Chap. 1 Vector Spaces (e) (f) (g) (h) (i) (j) (k) (1) If a vector space has a finite basis, then the number of vectors in every basis is the same. The dimension of P n (F) is n. The dimension of M m X n (F) is rn 4- n. Suppose that V is a finite-dimensional vector space, that Si is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then Si cannot contain more vectors than S2 . If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way. Every subspace of a finite-dimensional space is finite-dimensional. If V is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimen-sion n. If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent if and only if S spans V. 2. Determine which of the following sets are bases for R3. (a) {(1,0,-1), (2,5,1), (0 , -4 ,3)} (b) {(2,-4,1), (0,3,-1) , (6 ,0,-1)} (c) {(1,2,-1),(1,0,2),(2,1,1)} (d) {(-1,3,1), (2, - 4 , -3) , ( -3 ,8 ,2)} (e) {(1, - 3 , -2) , ( -3 ,1 ,3) , ( -2 , -10, -2 )} 3. Determine which of the following sets are bases for P2(R). (a) { - l - x + 2x2,2 + x - 2 x 2 , l - 2 x 4 - 4 x 2 } ( b ) { 1 4 - 2 X 4 - X 2 , 3 4 - X 2 , X 4 - X 2 } (c) {1 - 2x - 2x2, - 2 4- 3x - x 2 , 1 - x 4- 6x2} (d) { - 1 4- 2x 4- 4x2 ,3 - 4x - 10x2, - 2 - 5x - 6x2} (e) {1 4- 2x - x 2 ,4 - 2x 4- x2, - 1 4- 18x - 9x2} 4. Do the polynomials x3 2x24T,4x2 x4-3, and 3x 2 generate P,3(i?)? Justify your answer. 5. Is {(1,4, -6 ) , (1,5,8), (2,1,1), (0,1,0)} a linearly independent subset of R3? Justify your answer. 6. Give three different bases for F2 and for M2X2(F). 7. The vectors u : = (2,-3,1) , u2 = (1,4,-2) , u3 = ( -8 ,12 , -4 ) , uA = (1,37, 17), and u*, = (3, 5,8) generate R3. Find a subset of the set {ui,u2,U3,U4,ur,} that is a basis for R3. Sec. 1.6 Bases and Dimension 55 8. Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. The vectors wi = (2, -3 ,4 , -5 ,2) , u2 = ( -6 ,9 , -12,15, -6) , u3 = (3, -2 ,7 , -9 ,1) , u4 = (2, -8 ,2 , -2 ,6) , u5 = ( -1 ,1 ,2 ,1 , -3 ) , uQ = (0, - 3 , -18,9,12), u7 = (1,0, -2 ,3 , -2) , u8 = (2, -1 ,1 , -9 ,7 ) generate W. Find a subset of the set {ui, u2,... ,Ug} that is a basis for W. 9. The vectors m = (1,1,1,1), u2 = (0,1,1,1), u3 = (0,0,1,1), and W4 = (0,0,0,1) form a basis for F4. Find the unique representation of an arbitrary vector (01,02,03,04) in F4 as a linear combination of Mi, u2, U3, and U4. 10. In each part, use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the following points. (a) ( - 2 , - 6 ) , ( -1,5) , (1,3) (b) (-4,24), (1,9), (3,3) (c) (-2,3) , ( - 1 , - 6 ) , (1,0), (3 , -2) (d) ( -3 , -30 ) , ( -2,7) , (0,15), (1,10) / 11. Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and a and 6 are nonzero scalars, then both {u 4- v, au} and {au, bv} are also bases for V. 12. Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u 4- v 4- w, v + w, w} is also a basis for V. 13. The set of solutions to the system of linear equations xi - 2x2 4- x3 = 0 2xi 3x2 4- X3 = 0 is a subspace of R3. Find a basis for this subspace. 14. Find bases for the following subspaces of F5: Wi = {(ai ,a2,a3 ,a4,a5) F5: ax - a3 - a4 = 0} and W2 = {(ai, a2 , a3, a4 , a5) F5: a2 = a3 = a4 and ax 4- a 5 = 0}. What are the dimensions of Wi and W2? 56 Chap. 1 Vector Spaces 15. The set of all n x n matrices having trace equal to zero is a subspace W of M n x n (F ) (see Exam is the dimension of W? of M n x n (F ) (see Example 4 of Section 1.3). Find a basis for W. What 16. The set of all upper triangular n x n matrices is a subspace W of M n X n (F) (see Exercise 12 of Section 1.3). Find a basis for W. What is the dimension of W? 17. The set of all skew-symmetric n x n matrices is a subspace W of Mnxn(^) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W? 18. Find a basis for the vector space in Example 5 of Section 1.2. Justify your answer. 19. Complete the proof of Theorem 1.8. 20.* Let V be a vector space having dimension n, and let S be a subset of V that generates V. (a) Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.) (b) Prove that S contains at least n vectors. 21. Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset. 22. Let Wi and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and sufficient conditions on Wi and W2 so that d im(WinW 2 ) = dim(W1). 23. Let Vi,V2, ,vk,v be vectors in a vector space V, and define Wi = span({vi,v2,...,vk}), and W2 = span({vi,v2,... ,vk,v}). (a) Find necessary and sufficient conditions on v such that dim(Wi) = dim(W2). (b) State and prove a relationship involving dim(Wi) and dim(W2) in the case that dim(Wi) ^ dim(W2). 24. Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x) Pn(R) there exist scalars Co, c,...,cn such that g(x) = co/(x) 4- C l / ' (x ) 4- c2f'(x) + 4- cnf^(x), where f^n'(x) denotes the nth derivative of f(x). 25. Let V, W, and Z be as in Exercise 21 of Section 1.2. If V and W are vector spaces over F of dimensions m and n, determine the dimension ofZ. Sec. 1.6 Bases and Dimension 57 26. For a fixed a R, determine the dimension of the subspace of Pn(R) defined by {f Pn(R): f(a) = 0}. 27. Let Wi and W2 be the subspaces of P(F) defined in Exercise 25 in Section 1.3. Determine the dimensions of the subspaces Wi n P n (F) and W2 n Pn(F) . 28. Let V be a finite-dimensional vector space over C with dimension n. Prove that if V is now regarded as a vector space over R, then dim V = 2n. (See Examples 11 and 12.) Exercises 29-34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3. 29. (a) Prove that if W] and W2 are finite-dimensional subspaces of a vector space V, then the subspace Wi 4- W2 is finite-dimensional, and dim(Wi 4-W2) = dim(Wi) 4-dim(W2) - dim(W, n W2). Hint: Start with a basis {ui,u2,... ,uk} for Wi f) W2 and extend this set to a basis {ui, u2,..., uk,vi,v2,... vm} for Wi and to a basis {ui i2, . . . ,uk,wi,w2,... wp} for W2. (b) Let Wi and W2 be finite-dimensional subspaces of a vector space V, and let V = Wi 4- W2. Deduce that V is4he direct sum of Wi and W2 if and only if dim(V) = dim(Wi) 4- dim(W2). 30. Let V = M2X2(F), Wi = c a e V : a,b,cF and W2 = a /:a,bF Prove that W] and W2 are subspaces of V, and find the dimensions of Wi, W2, Wi 4- W2, and Wi n W2. 31. Let Wi and W2 be subspaces of a vector space V having dimensions rn and n, respectively, where m> n. (a) Prove that dim(Wi n W2) < n. (b) Prove that dim(Wi 4- W2) n > 0, such that dim(Wi D W2) = n. (b) Find an example of subspaces Wi and VV2 of R3 with dimensions m and n, where m > n > 0, such that dim(Wi 4- VV2) = m + n. 58 Chap. 1 Vector Spaces Find an example of subspaces Wi and W2 of R3 with dimensions m and n, where rn > n, such that both dim(Wi D W2) < n and dim(Wi 4- W2) < m 4-n. (c) 33. (a) Let Wi and W2 be subspaces of a vector space V such that V = Wi W2. If fti and ft2 are bases for Wi and W2, respectively, show that fti n k = 0 and fti U ft2 is a basis for V. (b) Conversely, let fti and ft2 be disjoint bases for subspaces Wi and W2, respectively, of a vector space V. Prove that if fti U ft2 is a basis for V, then V = Wi W2. 34. (a) Prove that if Wi is any subspace of a finite-dimensional vector space V, then there exists a subspace VV2 of V such that V = W i W 2 . (b) Let V = R2 and Wi = {(ai,0): ai R}. Give examples of two different subspaces W2 and W2 such that V = Wi W2 and V = WiW 2 . The following exercise requires familiarity with Exercise 31 of Section 1.3. 35. Let W be a subspace of a finite-dimensional vector space V, and consider the basis {ui,u2,... ,uk} for W. Let {ui,u2,... ,uk,uk+,... ,un} be an extension of this basis to a basis for V. (a) Prove that {uk+i + W, wfc+2 4- W , . . . , un 4- W} is a basis for V/W. (b) Derive a formula relating dim(V), dim(W), and dim(V/W). 1.7* MAXIMAL LINEARLY INDEPENDENT SUBSETS In this section, several significant results from Section 1.6 are extended to infinite-dimensional vector spaces. Our principal goal here is to prove that every vector space has a basis. This result is important in the study of infinite-dimensional vector spaces because it is often difficult to construct an explicit basis for such a space. Consider, for example, the vector space of real numbers over the field of rational numbers. There is no obvious way to construct a basis for this space, and yet it follows from the results of this section that such a basis does exist. The difficulty that arises in extending the theorems of the preceding sec-tion to infinite-dimensional vector spaces is that the principle of mathematical induction, which played a crucial role in many of the proofs of Section 1.6, is no longer adequate. Instead, a more general result called the maximal principle is needed. Before stating this principle, we need to introduce some terminology. Definition. Let J- be a family of sets. A member M of T is called maximal (with respect to set inclusion) if M is contained in no member of T other than M itself. Sec. 1.7 Maximal Linearly Independent Subsets 59 Example 1 Let T be the family of all subsets of a nonempty set S. (This family T is called the power set of S.) The set S is easily seen to be a maximal element oiT. Example 2 Let S and T be disjoint nonempty sets, and let T be the union of their power sets. Then S and T are both maximal elements of T. Example 3 Let T be the family of all finite subsets of an infinite set S. Then T has no maximal element. For if M is any member of T and s is any element of S that is not in M, then ML) {s} is a member of T that contains M as a proper subset. Definition. A collection of sets C is called a chain (or nest or tower) if for each pair of sets A and B in C, either A C B or B C A. Example 4 For each positive integer n let An = {1,2, . . . , n} . Then the collection of sets C = {An: n 1 ,2 ,3 , . . .} is a chain. In fact, Am C An if and only if m60 Chap. 1 Vector Spaces Example 5 Example 2 of Section 1.4 shows that {x3 - 2x2 - 5x - 3,3.x3 - 5a;2 - Ax - 9} is a maximal linearly independent subset of S = {2x3 - 2x2 + 2x - 6 ,x3 - 2x2 - hx - 3,3x3 - 5x2 - Ax - 9} in P2(-R). In this case, however, any subset of S consisting of two polynomials is easily shown to be a maximal linearly independent subset of S. Thus maximal linearly independent subsets of a set need not be unique. A basis ft for a vector space V is a maximal linearly independent subset of V, because 1. ft is linearly independent by definition. 2. If v V and v ^ ft, then ft U {v} is linearly dependent by Theorem 1.7 (p. 39) because span(/?) = V. Our next result shows that the converse of this statement is also true. Theorem 1.12. Let V be a vector space and S a subset that generates V. If ft is a maximal linearly independent subset ofS, then ft is a basis for V. Proof. Let ft be a maximal linearly independent subset of S. Because ft is linearly independent, it suffices to prove that ft generates V. We claim that S C span(/3), for otherwi.se there exists a v S such that v span(/?). Since Theorem 1.7 (p. 39) implies that ft U {v} is linearly independent, we have contradicted the maximality of ft. Therefore S C span(/3). Because span(S) = V, it follows from Theorem 1.5 (p. 30) that span(/3) = V. 0 Thus a subset of a vector space is a basis if and only if it is a maximal linearly independent subset of the vector space. Therefore we can accomplish our goal of proving that every vector space has a basis by showing that every vector space contains a maximal linearly independent subset. This result follows immediately from the next theorem. Theorem 1.13. Let S be a linearly independent subset of a vector space V. There exists a maximal linearly independent subset ofV that contains S. Proof. Let T denote the family of all linearly independent subsets of V that contain S. In order to show that T contains a maximal element, we must show that if C is a chain in T, then there exists a member U of T that contains each member of C. We claim that U, the union of the members of C, is the desired set. Clearly U contains each member of C, and so it suffices to prove Sec. 1.7 Maximal Linearly Independent Subsets 61 that U T (i.e., that U is a linearly independent subset, of V that contains S). Because each member of C is a subset of V containing S, we have S C U C V. Thus we need only prove that U is linearly independent. Let u, u2,... ,un be in U and a i , a 2 , . . . , an be scalars such that aiUi 4- a2w2 4- 4- anun = 0. Because Ui U for i 1 ,2 , . . . ,n, there exists a set Ai in C such that Ui Ai. But since C is a chain, one of these sets, say Ak, contains all the others. Thus ui Ak for i = 1,2, . . . , n . However, Ak is a linearly independent set; so aU + a2u2 4- 4- anun = 0 implies that a = a2 an = 0. It follows that U is linearly independent. The maximal principle implies that T has a maximal element. This el-ement is easily seen to be a maximal linearly independent subset of V that contains S. H Corollary. Every vector space has a basis. It can be shown, analogously to Corollary 1 of the replacement theorem (p. 46), that every basis for an infinite-dimensional vector space has the same cardinality. (Sets have the same cardinality if there is a one-to-one and onto mapping between them.) (See, for example, N. Jacobson, Lectures in Ab-stract Algebra, vol. 2, Linear Algebra. D. Van Nostrand Company, New York, 1953. p. 240.) Exercises 4-7 extend other results from Section 1.6 tri infinite-dimensional vector spaces. EXERCISES 1. Label the following statements as true or false. (a) Every family of sets contains a maximal element. (b) Every chain contains a maximal element. (c) If a family of sets has a maximal element, then that maximal element is unique. (d) If a chain of sets has a maximal clement, then that maximal ele-ment is unique. (e) A basis for a vector space is a maximal linearly independent subset of that vector space. (f) A maximal linearly independent subset of a vector space is a basis for that vector space. 2. Show that the set of convergent sequences is an infinite-dimensional subspace of the vector space of all sequences of real numbers. (See Exercise 21 in Section 1.3.) 3. Let V be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that V is infinite-dimensional. Hint: 62 Chap. 1 Vector Spaces Use the fact that IT is transcendental, that is, it is not a zero of any polynomial with rational coefficients. 4. Let W be a subspace of a (not necessarily finite-dimensional) vector space V. Prove that any basis for W is a subset of a basis for V. 5. Prove the following infinite-dimensional version of Theorem 1.8 (p. 43): Let ft be a subset of an infinite-dimensional vector space V. Then ft is a basis for V if and only if for each nonzero vector v in V, there exist unique vectors u, u2,..., un in ft and unique nonzero scalars C, c 2 , . . . , cn such that v ciui + c2u2 - 1- cnun. 6. Prove the following generalization of Theorem 1.9 (p. 44): Let Si and S2 be subsets of a vector space V such that Si C S2 . If Si is linearly independent and S2 generates V, then there exists a basis ft for V such that Si C ft C S2 . Hint: Apply the maximal principle to the family of all linearly independent subsets of S2 that contain Si, and proceed as in the proof of Theorem 1.13. 7. Prove the following generalization of the replacement theorem. Let ft be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset Si of ft such that S U Si is a basis forV. INDEX OF DEFINITIONS FOR CHAPTER 1 Additive inverse 12 Basis 43 Cancellation law 11 Column vector 8 Chain 59 Degree of a polynomial 9 Diagonal entries of a matrix Diagonal matrix 18 Dimension 47 Finite-dimensional space 46 Generates 30 Infinite-dimensional space 47 Lagrange interpolation formula Lagrange polynomials 52 Linear combination 24 Linearly dependent 36 Linearly independent 37 Matrix 8 Maximal element of a family of sets 58 52 Maximal linearly independent subset 59 n-tuple 7 Polynomial 9 Row vector 8 Scalar 7 Scalar multiplication 6 Sequence 11 Span of a subset 30 Spans 30 Square matrix 9 Standard basis for F' 43 Standard basis for Pn(F) 43 Subspace 16 Subspace generated by the elements of a set 30 Symmetric matrix 17 Trace 18 Transpose 17 Trivial representation of 0 36 Chap. 1 Index of Definitions Vector 7 Vector addition 6 Vector space 6 Zero matrix 8 63 Zero polynomial 9 Zero subspace 16 Zero vector 12 Zero vector space 15 / 2 L i n e a r T r a n s f o r m a t i o n s a n d M a t r i c e s 2.1 2.2 2.3 2.4 2.5 2.6* 2.7* Linear Transformations, Null spaces, and Ranges The Matrix Representation of a Linear Transformation Composition of Linear Transformations and Matrix Multiplication Invertibility and Isomorphisms The Change of Coordinate Matrix Dual Spaces Homogeneous Linear Differential Equations with Constant Coefficients i n Chapter 1, we developed the theory of abstract vector spaces in consid-erable detail. It is now natural to consider those functions defined on vector spaces that in some sense 'preserve' the structure. These special functions are called linear transformations, and they abound in both pure and applied mathematics. In calculus, the operations of differentiation and integration provide us with two of the most important examples of linear transforma-tions (see Examples 6 and 7 of Section 2.1). These two examples allow us to reformulate many of the problems in differential and integral equations in terms of linear transformations on particular vector spaces (see Sections 2.7 and 5.2). In geometry, rotations, reflections, and projections (see Examples 2, 3, and 4 of Section 2.1) provide us with another class of linear transformations. Later we use these transformations to study rigid motions in Rn (Section 6.10). In the remaining chapters, we see further examples of linear transforma-tions occurring in both the physical and the social sciences. Throughout this chapter, we assume that all vector spaces are over a common field F. 2.1 LINEAR TRANSFORMATIONS, NULL SPACES, AND RANGES In this section, we consider a number of examples of linear transformations. Many of these transformations are studied in more detail in later sections. Recall that a function T with domain V and codomain W is denoted by 64 Sec. 2.1 Linear Transformations, Null Spaces, and Ranges T: V - W. (See Appendix B.) 65 Definition. Let V and W be vector spaces (over F). We call a function T: V * W a linear transformation from V to W if, for all x, y V and c F, we have (a) T(x + y) = T(x) 4- T(y) and (b) T(cx) = cT(x). If the underlying field F is the field of rational numbers, then (a) implies (b) (see Exercise 37), but, in general (a) and (b) are logically independent. See Exercises 38 and 39. We often simply call T linear. The reader should verify the following properties of a function T: V W. (See Exercise 7.) 1. If T is linear, then T(0) = 0. 2. T is linear if and only if T(cx + y) = cT(x) + T(y) for all x, y V and cF. 3. If T is linear, then T(x -y) = T(x) - T(y) for all x. y V. 4. T is linear if and only if, for xi,x2.... .xn V and oi, o2,... , a F, we have T E ' > x < ) = X>T(-66 Chap. 2 Linear Transformations and Matrices T (01,02] (01,02] fa) Rotation (oi,a2] T(oi,fl2) (01,-02] (b) Reflection (01,02, T(ai,02) = (ai,0) (c) Projection Figure 2.1 As we will see in Chapter 6, the applications of linear algebra to geometry are wide and varied. The main reason for this is that most of the important geometrical transformations are linear. Three particular transformations that we now consider are rotation, reflection, and projection. We leave the proofs of linearity to the reader. Example 2 For any angle 0, define T#: R2 R2 by the rule: Te(ai,a2) is the vector obtained by rotating (ai ,a2) counterclockwise by 6 if (oi,a2) ^ (0,0), and T#(0,0) = (0,0). Then T#: R2 R2 is a linear transformation that is called the rotation by 0. We determine an explicit formula for T#. Fix a nonzero vector (ai ,a2) R2. Let a be the angle that (oi,a2) makes with the positive x-axis (see Figure 2.1(a)), and let r = ya2 + a2,. Then a,i rcosc* and a2 = rsino;. Also, Tff(ai,a2) has length r and makes an angle a + 9 with the positive x-axis. It follows that Tfl(ai,a2) = (rcos(a4-0),rsin(a!4-0)) = (r cos a cos 6 r sin a sin 0, r cos a sin 0 + r sin a cos 0) = (ai cos 9 a2 sin 0,ai sin 0 4- a2 cos 9). Finally, observe that this same formula is valid for (01,02) = (0,0). It is now easy to show, as in Example 1, that T# is linear. Example 3 Define T: R2 R2 by T(ai,o2) = (01,-02). T is called the reflection about the x -axis. (See Figure 2.1(b).) Example 4 Define T: R2 > R2 by T(oi,o2) = (oi,0). T is called the projection on the x-axis. (See Figure 2.1(c).) Sec. 2.1 Linear Transformations, Null Spaces, and Ranges 67 We now look at some additional examples of linear transformations. Example 5 Define T: Mmxn(-F) > M n x m ( F ) by T(A) = A1, where A1 is the transpose of A, defined in Section 1.3. Then T is a linear transformation by Exercise 3 of Section 1.3. Example 6 Define T: Pn(R) -* Pn-i(R) by T(/(x)) = f'(x), where f'(x) denotes the derivative of f(x). To show that T is linear, let g(x), h(x) Pn(R) and a R. Now T(ag(x) 4- h(x)) = (ag(x) + h(x))' = ag'(x) 4- h'(x) = aT(g(x)) + T(h(x)). So by property 2 above, T is linear. Example 7 Let V = C(R), the vector space of continuous real-valued functions on R. Let a,b R, a68 Example 8 Chap. 2 Linear Transformations and Matrices Let V and W be vector spaces, and let I: V > V and TQ: V > W be the identity and zero transformations, respectively. Then N(l) = {0}, R(l) = V, N(T0) = V, andR(To) = {0}. Example 9 Let T: R3 > R2 be the linear transformation defined by T(ai, 02,03) = (01 - a 2 , 2 o 3 ) . It is left as an exercise to verify that N(T) = {(a, a,0):aR} and R(T) = R2. In Examples 8 and 9, we see that the range and null space of each of the linear transformations is a subspace. The next result shows that this is true in general. Theorem 2.1. Let V and W be vector spaces and T: V -Then N(T) and R(T) are subspaces ofV and W, respectively. W be linear. Proof. To clarify the notation, we use the symbols Oy and 0w to denote the zero vectors of V and W, respectively. Since T(0V) = #w, we have that 0j N(T). Let x,y N(T) and c F. Then T(x + y) = T(x) + T(y) = 0yv + 0w = #w, and T(cx) = cT(x) = c0 w = #w- Hence x + y N(T) and ex N(T), so that N(T) is a subspace of V. Because T(0v) = 0N, we have that #w R(T). Now let x,y R(T) and c F. Then there exist v and w in V such that T(v) x and T(w) = y. So T(v 4- w) = T(v) 4- T(w) = x + y, and T(cv) = cT(v) ex. Thus x + y R(T) and ex R(T), so R(T) is a subspace of W. 1 The next theorem provides a method for finding a spanning set for the range of a linear transformation. With this accomplished, a basis for the range is easy to discover using the technique of Example 6 of Section 1.6. Theorem 2.2. Let V and W be vector spaces, and let T: V W be linear. If ft = {vi,v2,..., vn} is a basis for V, then R(T) = span(T(/?)) = span({T(Sec. 2.1 Linear Transformations, Null Spaces, and Ranges 69 Now suppose that w R(T). Then w T(v) for some G V . Because ft is a basis for V, we have n v = 2 J OiVi for some 01, a 2 , . . . , a n 6 F. i=i Since T is linear, it follows that n = T() = ^ a i T ^ ) span(T(/3)). 10 i=l So R(T) is contained in span(T (/?)). |J It should be noted that Theorem 2.2 is true if ft is infinite, that is. R(T) = Bpan.({T(v): v ft}). (See Exercise 33.) The next example illustrates the usefulness of Theorem 2.2. Example 10 Define the linear transformation T: P2(R) M2x2(R) by T(/(*) ) = / ( I ) - /(2) 0 0 /(0), Since ft {l,x,x2} is a basis for P2(R), we have R(T) = span(T(/5)) = span({T(l),T(,;),T(x2)}) = span = span 0 0 0 1 0 0 0 1 - 1 0 0 0 - 1 0 0 0 -3 0 0 0 Thus we have found a basis for R(T), and so dim(R(T)) = 2. As in Chapter 1, we measure the 'size' of a subspace by its dimension. The null space and range are so important that we attach special names to their respective dimensions. Definitions. Let V and W be vector spaces, and let T: V W be linear. If N(T) and R(T) are finite-dimensional, then we define the nullity ofT, denoted nullity(T), and the r ank of T, denoted rank(T), to be the dimensions of N(T) and R(T), respectively. Reflecting on the action of a linear transformation, we see intuitively that the larger the nullity, the smaller the rank. In other words, the more vectors that are carried into 0, the smaller the range. The same heuristic reasoning tells us that the larger the rank, the smaller the nullity. This balance between rank and nullity is made precise in the next theorem, appropriately called the dimension theorem. 70 Chap. 2 Linear Transformations and Matrices Theorem 2.3 (Dimension Theorem). Let V and W be vector spaces, and let T: V > W be linear. If V is finite-dimensional, then nullity(T) 4- rank(T) = dim(V). Proof. Suppose that dim(V) = n, dim(N(T)) = k, and {vi,v2,... ,vk} is a basis for N(T). By the corollary to Theorem 1.11 (p. 51), we may extend {vi,v2,... ,vk} to a basis ft = {vi,v2,... ,vn} for V. We claim that S = {T(vk+i), T(vk+2),..., T(vn)} is a basis for R(T). First we prove that S generates R(T). Using Theorem 2.2 and the fact that T(v-i) 0 for 1 < i < k, we have R(T) = s p a n ( { T M , T(v2),..., T(vn)} = span({T(v/c+i),T(v fc+2),..., T(vn)} = span(S). Now we prove that S is linearly independent. Suppose that n yi biT(vi) = 0 for 6fc+i, bk+2,..., bn F. i=fc+l Using the fact that T is linear, we have T ( J2 biVi) =-/ i=fc+i / So ^2 bivi e N(T)-i=k+l Hence there exist c,,c2,...,ck F such that n k k n ^2 biVi = y]ciVi or ^2(~Ci)vi + 22 ivi = -i=k+l i=l i=l i=k+l Since ft is a basis for V, we have bi 0 for all i. Hence S is linearly indepen-dent. Notice that this argument also shows that T(i>fc+i), T(vk+2),..., T(vn) are distinct; therefore rank(T) = n k. B If we apply the dimension theorem to the linear transformation T in Ex-ample 9, we have that nullity(T) 4- 2 = 3, so nullity(T) = 1. The reader should review the concepts of 'one-to-one' and 'onto' pre-sented in Appendix B. Interestingly, for a linear transformation, both of these concepts are intimately connected to the rank and nullity of the transforma-tion. This is demonstrated in the next two theorems. Sec. 2.1 Linear Transformations, Null Spaces, and Ranges 71 Theorem 2.4. Let V and W be vector spaces, and let T: V > W be linear. Then T is one-to-one if and only if N(T) = {0}. Proof. Suppose that T is one-to-one and x N(T). Then T(x) = 0 T(0). Since T is one-to-one, we have x 0. Hence N(T) = {0}. Now assume that N(T) = {0}, and suppose that T(x) = T(y). Then 0 = T(x) T(y) = T(x y) by property 3 on page 65. Therefore x y N(T) = {0}. So x y = 0, or x = y. This means that T is one-to-one. I The reader should observe that Theorem 2.4 allows us to conclude that the transformation defined in Example 9 is not one-to-one. Surprisingly, the conditions of one-to-one and onto are equivalent in an important special case. Theorem 2.5. Let V and W be vector spaces of equal (finite) dimension, and let T: V W be linear. Then the following are equivalent. (a) T is one-to-one. (b) T is onto. (c) rank(T)=dim(V). Proof. From the dimension theorem, we have nullity (T) 4- rank(T) = dim(V). Now, with the use of Theorem 2.4, we have that T is one-to-one if and only if N(T) = {0}, if and only if nullity(T) = 0, if and only if rank(T) = dim(V), if and only if rank(T) = dim(W), and if and only if dim(R(T)) = dim(W). By Theorem 1.11 (p. 50), this equality is equivalent to R(T) = W, the definition of T being onto. I We note that if V is not finite-dimensional and T: V V is linear, then it does not follow that one-to-one and onto are equivalent. (See Exercises 15, 16, and 21.) The linearity of T in Theorems 2.4 and 2.5 is essential, for it is easy to construct examples of functions from R into R that are not one-to-one, but are onto, and vice versa. The next two examples make use of the preceding theorems in determining whether a given linear transformation is one-to-one or onto. Example 11 Let T: P2(R) * P3(R) be the linear transformation defined by T(f(x)) = 2f'(x)-r fX3f(t)dt. Jo 72 Now Chap. 2 Linear Transformations and Matrices R(T) = span({T(l), T(x), T(x2)}) = span({3x, 2 4- -x2, Ax + x3}). Since {3x, 2 4- x2,Ax 4- x3} is linearly independent, rank(T) = 3. Since dim(P3(i?)) = 4, T is not onto. From the dimension theorem, nullity(T) + 3 = 3. So nullity(T) = 0, and therefore, N(T) = {0}. We conclude from Theorem 2.4 that T is one-to-one. Example 12 Let T : F2 + F2 be the linear transformation defined by T(oi,a2) = (oi +o 2 , o i ) . It is easy to see that N(T) = {0}; so T is one-to-one. Hence Theorem 2.5 tells us that T must be onto. In Exercise 14, it is stated that if T is linear and one-to-one, then a subset S is linearly independent if and only if T(S) is linearly independent. Example 13 illustrates the use of this result. Example 13 Let T: P2(R) R3 be the linear transformation defined by T(a0 4- a ix 4-a2a:2) = (ao,oi,o2). Clearly T is linear and one-to-one. Let S = {2 x + 3x2,x + x2,1 2.x2}. Then S is linearly independent in P2(i?) because T(S) = { ( 2 , - l , 3 ) , ( 0 , l , l ) , ( l , 0 , - 2 ) } is linearly independent in R3. In Example 13, we transferred a property from the vector space of polyno-mials to a property in the vector space of 3-tuples. This technique is exploited more fully later. One of the most important properties of a linear transformation is that it is completely determined by its action on a basis. This result, which follows from the next theorem and corollary, is used frequently throughout the book. Theorem 2.6. Let V and W be vector spaces over F, and suppose that {vi, u 2 , . . . , vn} is a basis for V. For w, w2,..., wn in W, there exists exactly one linear transformation T: V * W such that T(vi) = W{ for i = 1 ,2 , . . . , n. Sec. 2.1 Linear Transformations, Null Spaces, and Ranges Proof. Let x V. Then 73 X = y^QtTJj, i= where a i o 2 , . . . , o n are unique scalars. Define n T: V - W by T(x) = J ^ o ^ . i=i (a) T is linear: Suppose that u, v V and d F. Then we may write n n u y biVi and v = y CiVi i=l i=l for some scalars bi, b2,..., bn, ci, c2,..., cn. Thus du 4- v = ^2^bi + ('i)v>-i=1 So / T(du 4- v) y](dbj 4- c^w, = d y] bjWj 4- 22CiWj = dT(u) 4- T(v). i=i i=l i=l (b) Clearly T(vi) = ;, for ?' = 1 ,2 , . . . , n. (c) T is unique: Suppose that U: V * W is linear and U(t^) = Wi for i = 1 ,2 , . . . , n. Then for x V with = ^ O i U i , ?;=i have U(x) = y^QjUfoj) = J^OiWi = T(x). i=l i=l Hence U = T. I Corollary. Let V and W be vector spaces, and suppose that. V has a finite basis {vi, v2, , vn}. If U, T: V > W are linear and U(??,;) = T(f^) for i = 1 ,2 , . . . , n, then U = T. 74 Chap. 2 Linear Transformations and Matrices Example 14 Let T: R2 > R2 be the linear transformation defined by T(oi,o2) = (2a2 - o i , 3 a i ) , and suppose that U: R2 > R2 is linear. If we know that U(l,2) = (3,3) and U(l, 1) = (1,3), then U = T. This follows from the corollary and from the fact that {(1,2), (1,1)} is a basis for R2. EXERCISES 1. Label the following statements as true or false. In each part, V and W are finite-dimensional vector spaces (over F), and T is a function from V to W. (a) If T is linear, then T preserves sums and scalar products. (b) If T(x 4- y) = T(x) + T(y), then T is linear. (c) T is one-to-one if and only if the only vector x such that T(x) = 0 is x = 0. (d) If T is linear, then T(#v) = #w-(e) If T is linear, then nullity(T) 4- rank(T) = dim(W). (f) If T is linear, then T carries linearly independent subsets of V onto linearly independent subsets of W. (g) If T, U: V > W are both linear and agree on a basis for V, then T = U. (h) Given x i ,x 2 V and yi,y2 W, there exists a linear transforma-tion T: V > W such that T(xi) = yi and T(x2) = y2. For Exercises 2 through 6, prove that T is a linear transformation, and find bases for both N(T) and R(T). Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto. 2. T: R3 > R2 defined by T(oi,02,03) = (ai 02,203). 3. T: R2 -> R3 defined by T(a l 5a 2) = (01 4-o2,0,2o! - o 2 ) . 4. T: M 2 x 3 (F) - M 2 x 2 (F) defined by On 012 013 / '2an 012 ai3 + 2ai2 T 021 022 023 0 0 5. T: P2(R) - P3(R) defined by T(/(x)) = x/(x) 4- f'(x). Sec. 2.1 Linear Transformations, Null Spaces, and Ranges 75 6. T: M n x n ( F ) - F defined by T(A) = tr(A). Recall (Example 4, Sec-tion 1.3) that tv(A) = Y/Aii. 7. Prove properties 1, 2, 3, and 4 on page 65. 8. Prove that the transformations in Examples 2 and 3 are linear. 9. In this exercise, T: R2 > R2 is a function. For each of the following parts, state why T is not linear. (a) T(oi,o2) = (l ,o2) (b) T(ai ,a2) = (oi,of) (c) T(ai, a2) = (sinai,0) (d) T(oi,a2) = (|oi|,o2) (e) T(ai,a2) = (ai 4-1, a2) 10. Suppose that T: R2 -> R2 is linear, T(1,0) = (1,4), and T(l, 1) = (2,5). What is T(2,3)? Is T one-to-one? 11. Prove that there exists a linear transformation T: R2 > R3 such that T ( l , l ) = (1,0,2) and T(2,3) = (1,-1,4) . What / s T(8,11)? 12. Is there a linear transformation T: R3 R2 such that T(l, 0,3) = (1,1) and T ( - 2 , 0 , - 6 ) = (2,1)? 13. Let V and W be vector spaces, let T: V > W be linear, and let {w,it)2, ,wk} be a linearly independent subset of R(T). Prove that if S {vi,v2,... ,vk} is chosen so that T(vi) = Wi for i = 1 ,2 , . . . ,k , then S is linearly independent. 14. Let V and W be vector spaces and T: V > W be linear. (a) Prove that T is one-to-one if and only if T carries linearly inde-pendent subsets of V onto linearly independent subsets of W. (b) Suppose that T is one-to-one and that 5 is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly inde-pendent. (c) Suppose ft = {vi,v2,... ,vn} is a basis for V and T is one-to-one and onto. Prove that T(ft) = {T(vi),T(v2),... , T(?;Tl)} is a basis for W. 15. Recall the definition of P(R) on page 10. Define T : P ( J J ) - P ( J l ) by T(/(x)) = f(t)dt. Prove that T linear and one-to-one. but not onto. 76 Chap. 2 Linear Transformations and Matrices 16. Let T: P(R) -> P{R) be defined by T(/(x)) = f'(x). Recall that T is linear. Prove that T is onto, but not one-to-one. 17. Let V and W be finite-dimensional vector spaces and T: V linear. W be (a) Prove that if dim(V) < dim(W), then T cannot be onto. (b) Prove that if dim(V) > dim(W), then T cannot be one-to-one. 18. Give an example of a linear transformation T: R2 N(T) = R(T). R2 such that 19. Give an example of distinct linear transformations T and U such that N(T) = N(U) and R(T) = R(U). 20. Let V and W be vector spaces with subspaces Vi and Wi, respectively. If T: V W is linear, prove that. T(V() is a subspace of W and that {x V: T(x) Wi} is a subspace of V. 21. Let V be the vector space of sequences described in Example 5 of Sec-tion 1.2. Define the functions T, U: V - V by T(a i ,o 2 , . . . ) = (02,03,...) and U(ai ,a 2 , . . . ) = (0,ai ,a2 , ) T and U arc called the left shift and right shift operators on V, respectively. ^ (a) Prove that T and U are linear. (b) Prove that T is onto, but not one-to-one. (c) Prove that U is one-to-one, but not onto. 22. Let T: R3 R be linear. Show that there exist scalars a, b, and c such that T(x, y, z) = ax + by + cz for all (x, y, z) R3. Can you generalize this result for T: Fn F? State and prove an analogous result for T . pn . pra 23. Let T: R > R be linear. Describe geometrically the possibilities for the null space of T. Hint: Use Exercise 22. The following definition is used in Exercises 24 27 and in Exercise 30. Definition. Let V be a vector space and Wi and W2 be subspaces of V such that V = W] W2. (Recall the definition of direct sum given in the exercises of Section 1.3.) A function T: V > V is called the projection on Wi along VV2 if, for x = xi 4- x2 with xi Wi and x2 W2, we have T(x) = x. 24. Let T: R2 * R2. Include figures for each of the following parts. Sec. 2.1 Linear Transformations, Null Spaces, and Ranges 77 (a) Find a formula for T(a,b), where T represents the projection on the y-axis along the x-axis. (b) Find a formula for T(a, b), where T represents the projection on the y-axis along the line L {(s,s): s R}. 25. Let T: R: R3. (a) If T(o, b,c) = (a, 6,0), show that T is the projection on the xy-plane along the 2-axis. (b) Find a formula for T(a, b, c), where T represents the projection on the 2-axis along the xy-plane. (c) If T(a,b,c) = (a c, b, 0), show that T is the projection on the x;(/-plane along the line L = {(o, 0, a): a R}. 26. Using the notation in the definition above, assume that T: V > V is the projection on Wi along W2. (a) Prove that T is linear and W| = { J G V : T(X) = x}. (b) Prove that W, = R(T) and W2 = N(T). (c) Describe T if W, - V . (d) Describe T if Wi is the zero subspace. 27. Suppose that W is a subspace of a finite-dimensional vector space V. (a) Prove that there exists a subspace W and a function T: V > V such that T is a projection on W along W . (b) Give an example of a subspace W of a vector space V such that there are two projections on W along two (distinct) subspaces. The following definitions are used in Exercises 28 32. Definitions. Let V be a vector space, and let T: V V be linear. A subspace W of V is said to be T-invariant ifT(x) W for every x W, that is, T(W) C W. 7f W is T-iuvariaut, we define the restriction of T on W to be the function T w : W W defined by T w (x) = T(x) for all x W. Exercises 28-32 assume that W is a subspace of a vector space V and that T: V > V is linear. Warning: Do not assume that W is T-invariant or that T is a projection unless explicitly stated. 28. Prove that the subspaces {0}, V, R(T), and N(T) are all T-invariant. 29. If W is T-invariant, prove that Tw is linear. 30. Suppose that T is the projection on W along some subspace W . Prove that W is T-invariant and that Tw lw-31. Suppose that V = R(T)W and W is T-invariant. (Recall the definition of direct sum given in the exercises of Section 1.3.) 78 Chap. 2 Linear Transformations and Matrices (a) Prove that WCN(T) . (b) Show that if V is finite-dimensional, then W = N(T). (c) Show by example that the conclusion of (b) is not necessarily true if V is not finite-dimensional. 32. Suppose that W is T-invariant. Prove that N(TW) = N ( T ) n W and R(TW) = T(W). 33. Prove Theorem 2.2 for the case that ft is infinite, that is, R(T) = spm({T(v):vft}). 34. Prove the following generalization of Theorem 2.6: Let V and W be vector spaces over a common field, and let ft be a basis for V. Then for any function f:ft>N there exists exactly one linear transformation T: V -+ W such that T(x) = f(x) for all x ft. Exercises 35 and 36 assume the definition of direct sum given in the exercises of Section 1.3. 35. Let V be a finite-dimensional vector space and T: V V be linear. (a) Suppose that V = R(T) + N(T). Prove that V = R(T) N(T). (b) Suppose that R(T) n N(T) = {0}. Prove that V = R(T) N(T). Be careful to say in each part where finite-dimensionality is used. / 36. Let V and T be as defined in Exercise 21. (a) Prove that V = R(T) + N(T), but V is not a direct sum of these two spaces. Thus the result of Exercise 35(a) above cannot be proved without assuming that V is finite-dimensional. (b) Find a linear operator Tx on V such that R(TT) n N(Ti) = {0} but V is not a direct sum of R(Ti) and N(Ti). Conclude that V being finite-dimensional is also essential in Exercise 35(b). 37. A function T: V > W between vector spaces V and W is called additive if T(x + y) = T(x) 4- T(y) for all x, y V. Prove that if V and W are vector spaces over the field of rational numbers, then any additive function from V into W is a linear transformation. 38. Let T: C C be the function defined by T(z) = z. Prove that T is additive (as defined in Exercise 37) but not linear. 39. Prove that there is an additive function T: R > R (as defined in Ex-ercise 37) that is not linear. Hint: Let V be the set of real numbers regarded as a vector space over the field of rational numbers. By the corollary to Theorem 1.13 (p. 60), V has a basis ft. Let x and y be two distinct vectors in ft, and define / : ft > V by f(x) = y, f(y) = x, and f(z) = z otherwise. By Exercise 34, there exists a linear transformation Sec. 2.2 The Matrix Representation of a Linear Transformation 79 T: V -> V such that T(u) = f(u) for all u ft. Then T is additive, but for c = y/x, T(cx) ^ cT(x). The following exercise requires familiarity with the definition of quotient space given in Exercise 31 of Section 1.3. 40. Let V be a vector space and W be a subspace of V. Define the mapping n: V - V/W by jj(v) = v + W for v V. (a) Prove that 77 is a linear transformation from V onto V/W and that N(n) = W. (b) Suppose that V is finite-dimensional. Use (a) and the dimen-sion theorem to derive a formula relating dim(V), dim(W), and dim(V/W). (c) Read the proof of the dimension theorem. Compare the method of solving (b) with the method of deriving the same result as outlined in Exercise 35 of Section 1.6. 2.2 THE MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION Until now, we have studied linear transformations by examining their ranges and null spaces. In this section, we embark on one of the most useful ap-proaches to the analysis of a linear transformation on a finite-dimensional vector space: the representation of a linear transformation by a matrix. In fact, we develop a one-to-one correspondence between matrices and linear transformations that allows us to utilize properties of one to study properties of the other. We first need the concept of an ordered basis for a vector space. Definition. Let V be a finite-dimensional vector space. An ordered basis for V is a basis for V endowed with a specific order; that is, an ordered basis for V is a finite sequence of linearly independent vectors in V that generates V. Example 1 In F3, ft = {ei,e2,e3} can be considered an ordered basis. Also 7 = {e2, ei, 63} is an ordered basis, but ft ^ 7 as ordered bases. For the vector space Fn, we call {ei, e 2 , . . . , e n } the standard ordered basis for Fn. Similarly, for the vector space P n(F) , we call {1, x , . . . ,x n } the standard ordered basis for P n (F) . Now that we have the concept of ordered basis, we can identify abstract vectors in an n-dimensional vector space with n-tuples. This identification is provided through the use of coordinate vectors, as introduced next. 80 Chap. 2 Linear Transformations and Matrices Definition. Let ft {ui,u2,... ,un} be an ordered basis for a finite-dimensional vector space V. For x V, let oi, a 2 , . . . , a n be the unique scalars such that x = = y^QjMj. i=i We define the coordinate vector of x relative to ft, denoted [x]a, by / a i o2 x a = On/ Notice that [uip = ei in the preceding definition. It is left as an exercise to show that the correspondence x [x]/3 provides us with a linear transfor-mation from V to Fn. We study this transformation in Section 2.4 in more detail. Example 2 Let V = P2(R), and let ft = {1, x ,x 2} be the standard ordered basis for V. If f(x) = 4 + 6 x - 7 x 2 , then Let us now proceed with the promised matrix representation of a linear transformation. Suppose that V and W are finite-dimensional vector spaces with ordered bases ft = {v, v2, .. , vn} and 7 = {101,102, , wm}, respec-tively. Let T: V + W be linear. Then for each j, 1 < j < n, there exist unique scalars Oij F, 1 Sec. 2.2 The Matrix Representation of a Linear Transformation 81 Example 3 Let T: R2 * R3 be the linear transformation defined by T(ai, a2) = (oi 4- 3a2,0, 2oi - 4a2). Let ft and 7 be the standard ordered bases for R2 and R3, respectively. Now T(l, 0) = (1,0,2) = lei + 0e2 4- 2e3 and T(0,1) = (3,0, - 4 ) = 3d + 0e2 - 4e3. Hence If we let 7' = {e3, e2, ei}, then '2 - / m i ' = 1 0 0 1 . 1 3, Example 4 Let T: P3(-R) P2(-#) be the linear transformation defined by T(/(x)) = f'(x). Let ft and 7 be the standard ordered bases for P3(R) and P2(i?), respectively. Then T(l) = 0- l + 0 -x4-0 .x 2 T(x) = l-l-r-O-x + O-x2 T(x2) = 0 - l 4 - 2 - x + 0-x2 So T(x3) = 0 - l4 -0 -x4 -3 -x 2 . / 0 1 0 0' [T]} = 0 0 2 0 0 0 0 3, Note that when T(x3) is written as a linear combination of the vectors of 7, its coefficients give the entries of column j + 1 of [TH. 82 Chap. 2 Linear Transformations and Matrices Now that we have defined a procedure for associating matrices with linear transformations, we show in Theorem 2.8 that this association 'preserves' addition and scalar multiplication. To make this more explicit, we need some preliminary discussion about the addition and scalar multiplication of linear transformations. Definition. Let T, U: V W be arbitrary functions, where V and W are vector spaces over F, and let a F. We define T 4- U: V * W by (T + U)(x) = T(x) 4- U(x) for a/1 x V, and aT: V -> W by (aT)(x) = oT(x) for all x V. Of course, these are just the usual definitions of addition and scalar mul-tiplication of functions. We are fortunate, however, to have the result that both sums and scalar multiples of linear transformations are also linear. Theorem 2.7. Let V and W be vector spaces over a field F, and let T, U: V - W be linear. (a) For all a, F, aT 4- U is linear. (b) Using the operations of addition and scalar multiplication in the pre-ceding definition, the collection of all linear transformations from V to W is a vector space over F. Proof, (a) Let x, y V and c F. Then (aT 4- UWcx 4- y) = aT(cx 4- y) 4- U(cx 4- y) = a[T(cx + y)]4-cU(x) + U(y) = a[cT(x) + T(y)] + cU(x)4-U(y) = acT(x) 4- cU(x) 4- aT(y) + V(y) = c(aT + [)){x) + {aT + U){y). So aT 4- U is linear. (b) Noting that T0 , the zero transformation, plays the role of the zero vector, it is easy to verify that the axioms of a vector space are satisfied, and hence that the collection of all linear transformations from V into W is a vector space over F. II Definitions. Let V and W be vector spaces over F. We denote the vector space of all linear transformations from V into W by (V, W). In the case that V = W, we write (V) instead of (V, W). In Section 2.4, we see a complete identification of (V, W) with the vector space Mmxn(E), where n and rn are the dimensions of V and W, respectively. This identification is easily established by the use of the next theorem. Theorem 2.8. Let V and W be finite-dimensional vector spaces with ordered bases ft and 7, respectively, and let T, U: V > W be linear transfor-mations. Then Sec. 2.2 The Matrix Representation of a Linear Transformation 83 (a) [T + % = [T]} + M}and (b) [oT]I = a[T]^ for all scalars a. Proof. Let ft = {vi, v2,..., vn} and 7 = {wi,w2,... ,wm}. There exist unique scalars ay and />ij (1 < i < m, 1 < j < n) such that rn m T(t;j) = yjoij ' ; i a i l (l U(vj) = 7 ] 6jjt^j for 1 < j < n. j=] i=l Hence Thus (T + U K ^ ^ ^ O y + f c y J t B , . i= ( [ T 4 - U ] J ) i j = n , J 4 - ^ = ([T]54-[U]3)u. So (a) is proved, and the proof of (b) is similar. U Example 5 Let T: R2 > R3 and U: R2 > R3 be the linear transformations respectively defined by T(oi,a2) = (ai 4- 3a2,0.2oi 4a2) and U(oi,o2) = (ai - a2 ,2oi ,3ai 4- 2a2). Let ft and 7 be the standard ordered bases of R2 and R3, respectively. Then m j = [ 0 01 2 -A, (as computed in Example 3), and '1 - r U j 2 0 I .3 2, If we compute T 4- U using the preceding definitions, we obtain (T 4- U)(a,,a2) = (2ai 4- 2a2 ,2ai,5ai - 2a2). So [T + U ] J - [ 2 O j , which is simply [T]l + [U]^, illustrating Theorem 2.8. 84 Chap. 2 Linear Transformations and Matrices 1. 2. 4. 5. EXERCISES Label the following statements as true or false. Assume that V and W are finite-dimensional vector spaces with ordered bases ft and 7, respectively, and T, U: V > W are linear transformations. (a) For any scalar a, aT 4- U is a linear transformation from V to W. (b) [T]J = [[)]} implies that T = U. (c) If m = dim(V) and n = dim(W), then [T]^ is an m x n matrix. (d) [H-U]J = rT]J + [U]J-(e) (V, W) is a vector space. (f) (V,W) = (W,V). Let ft and 7 be the standard ordered bases for Rn and Rm , respectively. For each linear transformation T: Rn > Rm, compute [TTC. (a) (b) (c) (d) T:R2 T:R3 T:R3 T:R3 R3 defined by T(01,02) = (2ai a2,3ai 4-4a2,ai). R2 defined by T(ai, a2,03) = (2ai 4- 3a2 03, ai 4- 03). R defined by T(ai, 02,03) = 2ai 4- a2 303. R3 defined by (e) ( 0 (g) T: Rr T: Rr T: R7 T(ai,02,03) = (2a2 4 -03 , -0! 4-4a2 4-5a3 ,ai +03). -* Rn defined by T ( a i , a 2 , . . . , a n ) = ( a i , a i , . . . ,01). -* Rn defined by T (a i , a 2 , . . . ,a) = (o n , a _ i , . . . ,ai). -* R defined by T ( a i , a 2 , . . . , a n ) = ai 4- an. R3 be defined by T(ai ,a 2 ) = (ai a 2 , a i ,2a i 4-a2). Let ft be the standard ordered basis for R2 and 7 = {(1,1,0), (0,1,1), (2,2,3)}. Compute [T]J. If a = {(1,2), (2,3)}, compute [T]l. Define 3. Let T: R2 T: M2x2(R) Let 0 = P2(fl) by T = (a + b) + (2d)x 4- 6x2. and 7 = {l ,x,x2}. Compute [T]^. Let and ft={l,x,x1}, 7 = {1}-Sec. 2.2 The Matrix Representation of a Linear Transformation 85 (a) Define T: M 2 x 2 (F) -+ M 2 x 2 (F) by T(A) = AK Compute [T]a. (b) Define T: P2(R) - M2x2(R) by T(/(x)) = ^ 0 ) * j j j ) , where ' denotes differentiation. Compute [T]2. (c) Define T: M 2 x 2 (F) -> F by T(A) = tr(A). Compute [T]l. (d) Define T: P2(R) -* R by T(/(x)) = /(2). Compute [T]L (e) If A = 1 - 2 0 4 compute [i4]Q. (f) If f(x) = 3 6x 4- x2, compute [f(x)]@. (g) For a F, compute [a]7. 6. Complete the proof of part (b) of Theorem 2.7. 7. Prove part (b) of Theorem 2.8. / 8.* Let V be an n-dimensional vector space with an ordered basis ft. Define T: V > Fn by T(x) = [x]^. Prove that T is linear. 9. Let V be the vector space of complex numbers over the field R. Define T: V > V by T(z) = z, where z is the complex conjugate of z. Prove that T is linear, and compute [T]^, where ft {l,i}. (Recall by Exer-cise 38 of Section 2.1 that T is not linear if V is regarded as a vector space over the field C.) 10. Let V be a vector space with the ordered basis ft = {v,V2, , vn}. Define vo = 0. By Theorem 2.6 (p. 72), there exists a linear trans-formation T: V V such that T(VJ) = Vj 4- f j - i for j = 1 ,2 , . . . ,n. Compute [T]^. 11. Let V be an n-dimensional vector space, and let T: V * V be a linear transformation. Suppose that W is a T-invariant subspace of V (see the exercises of Section 2.1) having dimension k. Show that there is a basis ft for V such that [T]^ has the form A B O C where A is a k x k matrix and O is the (n k) x k zero matrix. 86 Chap. 2 Linear Transformations and Matrices 12. Let V be a finite-dimensional vector space and T be the projection on W along W , where W and W are subspaces of V. (See the definition in the exercises of Section 2.1 on page 76.) Find an ordered basis ft for V such that [T]^ is a diagonal matrix. 13. Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If R(T) n R(U) = {0}, prove that {T, U} is a linearly independent subset of (V, W). 14. Let V = P(R), and for j > 1 define T,-(/(a;)) = fu)(x), where f{j)(x) is the jib. derivative of f(x). Prove that the set {Ti, T 2 , . . . ,T n } is a linearly independent subset of (V) for any positive integer n. 15. Let V and W be vector spaces, and let S be a subset of V. Define 5 = {T (V,W): T(x) = 0 for all x S}. Prove the following statements. (a) 5 is a subspace of (V, W). (b) If Si and S2 are subsets of V and Si C S2, then S% C 5?. (c) If Vi and V2 are subspaces of V, then (Vi 4- V2) = V? n Vg. 16. Let V and W be vector spaces such that dim(V) = dim(W), and let T: V W be linear. Show that there exist ordered bases ft and 7 for V and W, respectively, such that [T]I is a diagonal matrix. 2.3 COMPOSITION OF LINEAR TRANSFORMATIONS AND MATRIX MULTIPLICATION In Section 2.2, we learned how to associate a matrix with a linear transforma-tion in such a way that both sums and scalar multiples of matrices are associ-ated with the corresponding sums and scalar multiples of the transformations. The question now arises as to how the matrix representation of a composite of linear transformations is related to the matrix representation of each of the associated linear transformations. The attempt to answer this question leads to a definition of matrix multiplication. We use the more convenient notation of UT rather than U o T for the composite of linear transformations U and T. (See Appendix B.) Our first result shows that the composite of linear transformations is lin-ear. Theo rem 2.9. Let V, W, and Z be vector spaces over the same field F, and let T: V -+ W and U: W - Z be linear. Then UT: V -> Z is linear. Proof. Let x, y V and a F. Then UT(ox 4- y) = U(T(ox 4- y)) = U(oT(x) 4- T(y)) = all(T(x)) 4- U(T(y)) = o(UT)(x) 4- UT(y). I Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 87 The following theorem lists some of the properties of the composition of linear transformations. Theorem 2.10. Let V be a vector space. Let T, Ui, U2 (V). Then (a) T(U, +U2 ) = TU i+TU 2 and(U] + U2)T = U1T4- U2T (b) T(UiU2) = (TU,)U2 (c) Tl = IT - T (d) a(UjU2) = (alli)U2 = U,(aU2) for all scalars a. Proof. Exercise. I A more general result holds for linear transformations that have domains unequal to their codomains. (See Exercise 8.) Let T: V W and U: W Z be linear transformations, and let A = [U]^ and B = [T], where a = {vi,v2,... ,vn}, ft = {ttfi,i02,...,u;m}, and 7 = {z,Z2, ,zp} are ordered bases for V, W, and Z, respectively. We would like to define the product AB of two matrices so that AB [UT]^. Consider the matrix [UT]^. For 1 < j < n, we have / m rn (UT)fe) = J(T(Vj)) = U ( ] Bkjwk = J2 BkMwk) ,fc=i k=l = E E ^ B kj i=l A;=1 i=l where Cij 2_, MkBkj-fe=i This computation motivates the following definition of matrix multiplication. Definition. Let A be an m x n matrix and B be an n x p matrix. We define the product of A and B, denoted AB, to be the m x p matrix such that n (AB)ij = Y, AikBkj for 1 < i < m, 1 < j < p. k= Note that (AB)ij is the sum of products of corresponding entries from the ith row of A and the j th column of B. Some interesting applications of this definition are presented at the end of this section. 88 Chap. 2 Linear Transformations and Matrices The reader should observe that in order for the product AB to be defined, there are restrictions regarding the relative sizes of A and B. The following mnemonic device is helpful: '(ra x n)'(n x p) = (ra x p)'; that is, in order for the product AB to be defined, the two 'inner' dimensions must be equal, and the two 'outer' dimensions yield the size of the product. Example 1 We have 1-44-2-24-1-5 0 - 4 4 - 4 - 2 4 - ( - l ) - 5 Notice again the symbolic relationship ( 2 x 3 ) - ( 3 x l) = 2 x 1. As in the case with composition of functions, we have that matrix multi-plication is not commutative. Consider the following two products: 1 1 0 0 0 1 1 0 1 1 0 0 and 0 1 1 0 1 1 0 0 0 0 1 1 Hence we see that even if both of the matrix products AB and BA are defined, it need not be true that AB = BA. Recalling the definition of the transpose of a matrix from Section 1.3, we show that if A is an mxn matrix and B is an nxp matrix, then (AB)1 = BtAt. Since (AB)tij = (AB)ji = J2AjkBki fc=i and [B*A% = ] ( B * ) ( A % = J2BkiA^ fc=i fc=i we are finished. Therefore the transpose of a product is the product of the transposes in the opposite order. The next theorem is an immediate consequence of our definition of matrix multiplication. Theorem 2.11. Let V, W, and Z be finite-dimensional vector spaces with ordered bases a, ft, and 7, respectively. Let T: V * W and U: W * Z be linear transformations. Then [UTE = [urjmS-Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 89 Corollary. Let V be a finite-dimensional vector space with an ordered basis ft. Let T,U (V). Then [[)T]0 = [V]p[T]p. We illustrate Theorem 2.11 in the next example. Example 2 Let U:'P3(i2) -> P2(R) and T: P2(R) -* ?3{R) be the linear transformations respectively defined by U(/(x)) = f'(x) and T(/(x)) = f f(t) dt. Jo Let a and /? be the standard ordered bases of Ps(R) and P2(R), respectively. From calculus, it follows that UT = I, the identity transformation on P2(R). To illustrate Theorem 2.11, observe that '0 1 0 0N / 1 0 0 0 1 2 0 o 0 0 1 3 / [UTl/j = [UlSm? = | 0 0 2 0 .0 0 0 3, The preceding 3 x 3 diagonal matrix is called an identity matrix and i defined next, along with a very useful notation, the Kronecker delta. is Definitions. We define the Kronecker delta 6ij by Sij = lifi=j and 6ij =0ifi^j. The n x n identity matrix In is defined by (In)ij Oy. Thus, for example, / i = (l), h = 1 0 0 1 and 1% The next theorem provides analogs of (a), (c), and (d) of Theorem 2.10. Theorem 2.10(b) has its analog in Theorem 2.16. Observe also that part (c) of the next theorem illustrates that the identity matrix acts as a multiplicative identity in M n X n (F) . When the context is clear, we sometimes omit the subscript n from In. Theorem 2.12. Let A be an m x n matrix, B and C be n x p matrices, and D and E be q x rn matrices. Then (a) A(B -t-C) = AB + AC and (D 4- E)A = DA + EA. (b) a(AB) = (aA)B A(aB) for any scalar a. (c) ImA = A = AIn. (d) If V is an n-dimensional vector space with an ordered basis ft, then [lv]/3 = In-90 Chap. 2 Linear Transformations and Matrices Proof. We prove the first half of (a) and (c) and leave the remaining proofs as an exercise. (See Exercise 5.) (a) We have [A(B + C)}ij = ] T Aik(B 4- C)kj = J2 Aik(Bkj + Ckj) k=i fc=i n n n = ^2(AikBkj 4- AikCkj) = J2 AikBkj 4- Y AikCkj k= k=l fc=l = (AB)ij 4- (AC)ij = [AB 4- AC]y. So A(B + C) = AB + AC. (c) We have m m [ImA)ij = y ^lrn)ikAkj / Ojfc-^ fcj = Ay. fc=l fc=l Corollary. Let A be an ra x n matrix, Bi,B2,- ,Bk benxp matrices, Ci, C 2 , . . . , Cfc be qxm matrices, and ai,a2,... ,ak be scalars. Then Al^OiBi) =YjaiABi j=i i= and / < fc fc Y aiCi ) A = X ] OiCi A vi=I / i=l Proof. Exercise. For an n x n matrix A, we define A.1 = A, A2 = AA, A3 A2A, and, in general, Ak = Ak~l A for k = 2 ,3 , . . . . We define A0 = In. With this notation, we see that if A = 0 0 1 0 / ' then A2 = O (the zero matrix) even though A ^ O. Thus the cancellation property for multiplication in fields is not valid for matrices. To see why, assume that the cancellation law is valid. Then, from A-A = A2 = O = AO, we would conclude that A O, which is false. Theorem 2.13. Let A be an ra x n matrix and B be an n x p matrix. For each j (1 < j < p) let Uj and Vj denote the jth columns of AB and B, respectively. Then Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 91 (a) Uj = AVJ (b) Vj BCJ, where ej is the jth standard vector of Fp. Proof, (a) We have f(AB)u (AB)2j {AB)mj) fc=i ^2A2kBk kj fc=l 2 4 AmkBkj Vfc=i = A (Bij B2j {BnjJ = Av J-Hence (a) is proved. The proof of (b) is left as an exercise. (See Exercise 6.) It follows (sec Exercise 14) from Theorem 2.13 that column j of AB is a linear combination of the columns of A with the coefficients in the linear combination being the entries of column j of B. An analogous result holds for rows; that is, row i of AB is a linear combination of the rows of B with the coefficients in the linear combination being the entries of row i of A. The next result justifies much of our past work. It utilizes both the matrix representation of a linear transformation and matrix multiplication in order to evaluate the transformation at any given vector. Theorem 2.14. Let V and W be finite-dimensional vector spaces having ordered bases ft and 7, respectively, and let T: V W be linear. Then, for each u V, we have [T(u)]7 = [T]}[u}0. Proof. Fix u V, and define the linear transformations / : F * V by /(a) = au and g: F W by g(a) = aT(u) for all a F. Let a = {1} be the standard ordered basis for F. Notice that g = If. Identifying column vectors as matrices and using Theorem 2.11, we obtain [T(ti)]7 = mh = M = PVE = mj i /12 = m j [ / ( i ) ] / j = m j M / i - Example 3 Let T: Ps(i?) P 2(#) be the linear transformation defined by T(/(x)) = f'(x), and let ft and 7 be the standard ordered bases for Ps(R) and P2(i?,), respectively. If A [T]L then, from Example 4 of Section 2.2, we have /0 1 0 0N A = I 0 0 2 0 0 0 0 3, 92 Chap. 2 Linear Transformations and Matrices We illustrate Theorem 2.14 by verifying that [T(p(x))]7 = [T]l[p(x)]/3, where p(x) P3(R) is the polynomial p(x) = 2 4x4-x24-3x3. Let q(x) T(p(x)); then q(x) = p'(x) = A + 2x 4- 9x2. Hence ;T(p(.r)) . !,/(,-)], | 2 but also raw* = AP(X)]0 = '0 1 0 0' 0 0 2 0 0 0 0 3, ( 2 -A 1 V V We complete this section with the introduction of the left-multiplication transformation LA, where A is an mxn matrix. This transformation is proba-bly the most important tool for transferring properties about transformations to analogous properties about matrices and vice versa. For example, we use it to prove that matrix multiplication is associative. Definition. Let A be an ra x n matrix with entries from a field F. We denote by LA the mapping LA- F' * FTO defined by L^(x) = Ax (the matrix product of A and x) for each column vector x Fn. We call LA a left-multiplication transformation. Example 4 Let A = 1 2 1 0 1 2 Then A M2x3(R) and L^: R3 -> R2. If then LA(x) = Ax = 1 2 0 1 We see in the next theorem that not only is L^ linear, but, in fact, it has a great many other useful properties. These properties are all quite natural and so are easy to remember. Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 93 LB]} = B. Theorem 2.15. Let A be an m x n matrix with entries from F. Then the left-multiplication transformation LA '- Fn > Fm is linear. Furthermore, if B is any other ra x n matrix (with entries from F) and ft and 7 are the standard ordered bases for Fn and Fm, respectively, then we have the following properties. (a) [LA]} = A. (b) LA = LB if and only if A = B. (c) LA+B = LA 4- LB a n d LaA = OLA for all a F. (d) If T: Fn * Fm is linear, then there exists a uniquemxn matrixC such that T = Lc- In fact, C = [T]}. (e) If E is an n x p matrix, then LAE = ^-ALE-(f) Ifm = n, then Ljn = lp. Proof. The fact that LA is linear follows immediately from Theorem 2.12. (a) The jth column of [LA]} is equal to LA(CJ). However L^e^) = Aej, which is also the jth column of A by Theorem 2.13(b). So [LA]} = A. (b) If LA = LB, then we may use (a) to write A [LA]} Hence A B. The proof of the converse is trivial. (c) The proof is left as an exercise. (See Exercise 7.) (d) Let C = [T]}. By Theorem 2.14, we have [T(x)]7 = [ T ^ x ] ^ or T(x) = Cx = Lc(x) for all x F n . So T = L^. The uniqueness of C follows from (b). (e) For any j (1 < j < p), we may apply Theorem 2.13 several times to note that (AE)ej is the jth column of AE and that the jth column of AE is also equal to A(Eej). So (AE)ej = A(Ee,j). Thus LAE(ej) = (AE)ej = A(Eej) = LA(Ee5) = LA(LB(e,)). Hence LAE = L^Ls by the corollary to Theorem 2.6 (p. 73). (f) The proof is left as an exercise. (See Exercise 7.) 1 We now use left-multiplication transformations to establish the associa-tivity of matrix multiplication. Theorem 2.16. Let A,B, and C be matrices such that A(BC) is de-fined. Then (AB)C is also defined and A(BC) = (AB)C; that is, matrix multiplication is associative. Proof. It is left to the reader to show that (AB)C is defined. Using (e) of Theorem 2.15 and the associativity of functional composition (see Ap-pendix B), we have LA(BC) = LALBC = LA(LBLC) ( L ^ L B ) L C = L^BI-C = L^AB)C-So from (b) of Theorem 2.15, it follows that A(BC) = (AB)C. | 94 Chap. 2 Linear Transformations and Matrices Needless to say, this theorem could be proved directly from the definition of matrix multiplication (see Exercise 18). The proof above, however, provides a prototype of many of the arguments that utilize the relationships between linear transformations and matrices. Applications A large and varied collection of interesting applications arises in connec-tion with special matrices called incidence matrices. An incidence matrix is a square matrix in which all the entries are either zero or one and, for convenience, all the diagonal entries are zero. If we have a relationship on a set of n objects that we denote by 1,2, . . . ,n, then we define the associated incidence matrix A by Ay = 1 if i is related to j, and Ay = 0 otherwise. To make things concrete, suppose that we have four people, each of whom owns a communication device. If the relationship on this group is 'can trans-mit to,' then Ay = 1 if i can send a message to j, and Ay = 0 otherwise. Suppose that A = fO 1 0 0 1 0 0 1 0 1 0 1 1 1 0 0 / Then since A34 = 1 and A14 = 0, we see that person 3 can send to 4 but 1 cannot send to 4. We obtain an interesting interpretation of the entries of A2. Consider, for instance, (A2)3I = A31A11 4- A32A21 4- A33A31 4- A34A41. Note that any term A3fcAfci equals 1 if and only if both A3fc and Afci equal 1, that is, if and only if 3 can send to k and k can send to 1. Thus (A2)3i gives the number of ways in which 3 can send to 1 in two stages (or in one relay). Since A2 = ( 0 0 1 1 2 0 0 2 1 0 1 Vi 1 0 17 we see that there are two ways 3 can send to 1 in two stages. In general, (A + A2 + 4- Am )y is the number of ways in which i can send to j in at most ra stages. A maximal collection of three or more people with the property that any two can send to each other is called a clique. The problem of determining cliques is difficult, but there is a simple method for determining if someone Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 95 belongs to a clique. If we define a new matrix B by By = 1 if i and j can send to each other, and By = 0 otherwise, then it can be shown (see Exercise 19) that person i belongs to a clique if and only if (B3)u > 0. For example, suppose that the incidence matrix associated with some relationship is A = To determine which people belong to cliques, we form the matrix B, described earlier, and compute B3 . In this case, / o 1 1 1 1 0 1 0 1 0 1 0 1 1 1 0 / B = (0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 o and B 3 = (0 A A 0 0 4 U 0 0 4 0 4 4 0 4 0 Since all the diagonal entries of B 3 are zero, we conclude that there are no cliques in this relationship. Our final example of the use of incidence matrices is concerned with the concept of dominance. A relation among a group of people is called a dom-inance relation if the associated incidence matrix A has the property that for all distinct pairs i and j, Aij = 1 if and only if Aji 0, that is, given any two people, exactly one of them dominates (or, using the terminology of our first example, can send a message to) the other. Since A is an incidence matrix, An = 0 for all i. For such a relation, it can be shown (see Exercise 21) that the matrix A 4- A2 has a row [column] in which each entry is positive except for the diagonal entry. In other words, there is at least one person who dominates [is dominated by] all others in one or two stages. In fact, it can be shown that any person who dominates [is dominated by] the greatest number of people in the first stage has this property. Consider, for example, the matrix A = The reader should verify that this matrix corresponds to a dominance relation. Now / o 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 0 0 0 0 1 o^ A + A2 = (0 1 1 1 V2 2 0 2 2 2 1 1 0 2 2 1 1 2 0 2 1 0 1 1 V 96 Chap. 2 Linear Transformations and Matrices Thus persons 1, 3, 4, and 5 dominate (can send messages to) all the others in at most two stages, while persons 1, 2, 3, and 4 are dominated by (can receive messages from) all the others in at most two stages. EXERCISES 1. Label the following statements as true or false. In each part, V,W, and Z denote vector spaces with ordered (finite) bases a, ft, and 7, respectively; T: V W and U: W * Z denote linear transformations; and A and B denote matrices. (a) [UTfc - mg[U]?. (b) T(v)]0 = mSMa for all v V. (c) [U(w)]p = [J]P[w]l3 for ail wN. (d) [lv] = / . (e) [T2]g = ([T]g)2. (f) A2 = I implies that A = I or A = -I. (g) T = LA for some matrix A. (h) A2 = O implies that A = O, where O denotes the zero matrix, (i) L ^ + B = LA + LB. (j) If A is square and Ay = Sec. 2.3 Composition of Linear Transformations and Matrix Multiplication 97 (a) Compute [[)]}, [T]^, and [UTg directly. Then use Theorem 2.11 to verify your result. (b) Let h(x) = 3 - 2x 4- x2. Compute [h{x)]a and [U(/t(x))]7. Then use [[)]} from (a) and Theorem 2.14 to verify your result. 4. For each of the following parts, let T be the linear transformation defined in the corresponding part of Exercise 5 of Section 2.2. Use Theorem 2.14 to compute the following vectors: (a) [T(A)]a, where A = ( * (b) [ T ( / ( x ) ) ] a , w h e r e / ( . r ) = 4 -(c) [T(A)]7, where A=(l f (d) [T(/(x))]7, where f(x) = 6 - x + 2x2. 5. Complete the proof of Theorem 2.12 and its corollary. 6. Prove (b) of Theorem 2.13. 7. Prove (c) and (f) of Theorem 2.15. 8. Prove Theorem 2.10. Now state and prove a more general result involv-ing linear transformations with domains unequal/to their codomains. 9. Find linear transformations U,T: F2 F2 such that UT = To (the zero transformation) but TU ^ To- Use your answer to find matrices A and B such that AB = O but BA^O. 10. Let A be an n x n matrix. Prove that A is a diagonal matrix if and only if Ay = SijAjj for all i and j. 11. Let V be a vector space, and let T: V > V be linear. Prove that T2 = T() if and only if R(T) C N(T). 12. Let V, W. and Z be vector spaces, and let T: V W and U: W Z be linear. (a) Prove that if UT is one-to-one, then T is one-to-one. Must U also be one-to-one? (b) Prove that if UT is onto, then U is onto. Must T also be onto? (c) Prove that if U and T are one-to-one and onto, then UT is also. 13. Let A and B be n x n matrices. Recall that the trace of A is defined by tr(A) = ) A i i . *=i Prove that tr(AB) = tr(BA) and tr(A) = tr(A'). 98 Chap. 2 Linear Transformations and Matrices 14. Assume the notation in Theorem 2.13. (a) Suppose that z is a (column) vector in Fp. Use Theorem 2.13(b) to prove that Bz is a linear combination of the columns of B. In particular, if z = (ai,a2,..., ap)1, then show that Bz = J2a i=l JUJ-(b) (c) (d) Extend (a) to prove that column j of AB is a linear combination of the columns of A with the coefficients in the linear combination being the entries of column j of B. For any row vector w Fm, prove that wA is a linear combination of the rows of A with the coefficients in the linear combination being the coordinates of w. Hint: Use properties of the transpose operation applied to (a). Prove the analogous result to (b) about rows: Row i of AB is a linear combination of the rows of B with the coefficients in the linear combination being the entries of row i of A. 15. * Let M and A be matrices for which the product matrix MA is defined. If the j th column of A is a linear combination of a set of columns of A, prove that the jth column of MA is a linear combination of the corresponding coftimns of MA with the same corresponding coefficients. 16. Let V be a finite-dimensional vector space, and let T: V V be linear. (a) If rank(T) = rank(T2), prove that R(T) n N(f) = {0}. Deduce that V = R(T) e N(T) (see the exercises of Section 1.3). (b) Prove that V = R(Tfc) N(Tfc) for some positive integer k. 17. Let V be a vector space. Determine all linear transformations T: V * V such that T = T2. Hint: Note that x = T(x) 4- (x - T(x)) for every x in V, and show that V = {y: T(y) = y} N(T) (see the exercises of Section 1.3). 18. Using only the definition of matrix multiplication, prove that multipli-cation of matrices is associative. 19. For an incidence matrix A with related matrix B defined by By = 1 if i is related to j and j is related to i, and By = 0 otherwise, prove that i belongs to a clique if and only if (B3)u > 0. 20. Use Exercise 19 to determine the cliques in the relations corresponding to the following incidence matrices. Sec. 2.4 Invertibility and Isomorphisms 99 (a) (() 1 0 1 1 0 0 0 0 1 0 1 l 0 1 ()) (b) {{) 0 1 1 1 0 0 1 1 0 0 1 1 0 1 0 / 21. Let A be an incidence matrix that is associated with a dominance rela-tion. Prove that the matrix A 4- A2 has a row [column] in which each entry is positive except for the diagonal entry. 22. Prove that the matrix A = corresponds to a dominance relation. Use Exercise 21 to determine which persons dominate [are dominated by] each of the others within two stages. 23. Let A be an n x n incidence matrix that corresponds to a dominance relation. Determine the number of nonzero entries of A. / 2.4 INVERTIBILITY AND ISOMORPHISMS The concept of invertibility is introduced quite early in the study of functions. Fortunately, many of the intrinsic properties of functions are shared by their inverses. For example, in calculus we learn that the properties of being con-tinuous or differentiable are generally retained by the inverse functions. We see in this section (Theorem 2.17) that the inverse of a linear transformation is also linear. This result greatly aids us in the study of inverses of matrices. As one might expect from Section 2.3, the inverse of the left-multiplication transformation L,. (when it exists) can be used to determine properties of the inverse of the matrix A. In the remainder of this section, we apply many of the results about in-vertibility to the concept of isomorphism. We will see that finite-dimensional vector spaces (over F) of equal dimension may be identified. These ideas will be made precise shortly. The facts about inverse functions presented in Appendix B are, of course, true for linear transformations. Nevertheless, we repeat some of the defini-tions for use in this section. Definition. Let V and W be vector spaces, and let T: V > W be linear. A function U: W V is said to be an inverse of T if TU lw and UT = lv-If T has an inverse, then T is said to be invertible. As noted in Appendix B, if T is invertible. then the inverse of T is unique and is denoted by T~l. 100 Chap. 2 Linear Transformations and Matrices The following facts hold for invertible functions T and U. 1. (TU)-1 = U~1T-1 . 2. ( T - 1 ) - 1 = T; in particular, T _ 1 is invertible. We often use the fact that a function is invertible if and only if it is both one-to-one and onto. We can therefore restate Theorem 2.5 as follows. 3. Let T: V W be a linear transformation, where V and W are finite-dimensional spaces of equal dimension. Then T is invertible if and only if rank(T) = dim(V). Example 1 Let T: Pi(B) R2 be the linear transformation defined by T(a 4- bx) = (a, a 4- b). The reader can verify directly that T _ 1 : R2 > Pi(B) is defined by T_ 1(c, d) = c + (d c)x. Observe that T _ 1 is also linear. As Theorem 2.17 demonstrates, this is true in general. Theorem 2.17. Let V and W be vector spaces, and let T: V linear and invertible. Then T _ 1 : W > V is linear. W be Proof. Let 3/1,2/2 W and c F. Since T is onto and one-to-one, there exist unique vectors xi and x2 such that T(xi) = yi and T(x2) = j/2. Thus xi - T-^ j / i ) and x2 = T~1(y2); so f T - ^ q / ! 4- y2) = T-1[cT(xi) 4- T(x2)] = J-l[T(cxi 4- x2)] = c x i - r x 2 = cT-1(j/1)4-T-1(j/2). | It now follows immediately from Theorem 2.5 (p. 71) that if T is a linear transformation between vector spaces of equal (finite) dimension, then the conditions of being invertible, one-to-one, and onto are all equivalent. We are now ready to define the inverse of a matrix. The reader should note the analogy with the inverse of a linear transformation. Definition. Let A be an n x n matrix. Then A is invertible if there exists annx n matrix B such that AB = BA I. If A is invertible, then the matrix B such that AB = BA = J is unique. (If C were another such matrix, then C = CI - C(AB) = (CA)B = IB = B.) The matrix B is called the inverse of A and is denoted by A - 1 . Example 2 The reader should verify that the inverse of 5 7 2 3 is 3 - 7 - 2 5 Sec. 2.4 Invertibility and Isomorphisms 101 In Section 3.2, we learn a technique for computing the inverse of a matrix. At this point, we develop a number of results that relate the inverses of matrices to the inverses of linear transformations. Lemma. Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, dim(V) = dim(W). Proof. Suppose that V is finite-dimensional. Let ft = {x,X2, , xn} be a basis for V. By Theorem 2.2 (p. 68), T(/?) spans R(T) = W; hence W is finite-dimensional by Theorem 1.9 (p. 44). Conversely, if W is finite-dimensional, then so is V by a similar argument, using T _ 1 . Now suppose that V and W are finite-dimensional. Because T is one-to-one and onto, we have nullity(T) = 0 and rank(T) = dim(R(T)) = dim(W). So by the dimension theorem (p. 70), it follows that dim(V) = dim(W). 1 Theorem 2.18. Let V and W be finite-dimensional vector spaces with ordered bases (3 and 7, respectively. Let T: V > W be linear. Then T is invertible if and only if [T]^ is invertible. Furthermore, ^T-1]^ = ([T]2)_1. Proof. Suppose that T is invertible. By the lemma, we have dim(V) = dim(W). Let n = dim(V). So [T]l is an n x n matrix. Now T _ 1 : W V satisfies TT _ 1 = lw and T - 1 T = lv- Thus In = [Ivfe = [T- l Tl , = [ T - 1 ] ^ . Similarly, [ T g p - 1 ] ? = Jn . So [T]} is invertible and ( [ T g ) ' 1 = [T'1]^. Now suppose that A = [T]Z is invertible. Then there exists an n x n matrix B such that AB = BA = In. By Theorem 2.6 (p. 72), there exists U e ( W , V ) such that u(wj) = ^2 BiiVi fori = 1 ,2 , . . . , n, i=l where 7 = {101,142,..., wn} and 0 = {i, ifc,..., vn}. It follows that [U]^ = B. To show that U = T _ 1 , observe that [iJT]0 = [^[T}} = BA = In = {W}0 by Theorem 2.11 (p. 88). So UT = lv , and similarly, TU = lw . I 102 Chap. 2 Linear Transformations and Matrices E x a m p l e 3 Let 0 and 7 be the standard ordered bases of Pi(R) and R2, respectively. For T as in Example 1, we have TO=(i ' ' - ' ] ? = ( _ ; ; It can be verified by matrix multiplication that each matrix is the inverse of the other. Corollary 1. Let V be a finite-dimensional vector space with an ordered basis (3, and let T: V > V be linear. Then T is invertible if and only if [T]# is invertible. Furthermore, [T - 1]^ = (PI/3) - . Proof. Exercise. !j Corollary 2. Let A be ann x n matrix. Then A is invertible if and only if-A is invertible. Furthermore, (La) - 1 = I-A-'-Proof. Exercise. @ The notion of invertibility may be used to formalize what may already have been observed by the reader, that is, that certain vector spaces strongly resemble one another except for the form of their vectors. For example, in the case of M2x2(^) and F4, if we associate to each matrix a 6 c d the 4-tuple (a, 6, c, d), we see that sums and scalar products associate in a similar manner; that is, in terms of the vector space structure, these two vector spaces may be considered identical or isomorphic. Definitions. Let V and W be vector spaces. We say that V is isomor-phic to W if there exists a linear transformation T: V > W that is invertible. Such a linear transformation is called an isomorphism from V onto W. We leave as an exercise (see Exercise 13) the proof that 'is isomorphic to' is an equivalence relation. (See Appendix A.) So we need only say that V and W are isomorphic. E x a m p l e 4 Define T: F2 * P.i(F) by T(a,a2) = fli + nix. It is easily checked that T is an isomorphism; so F2 is isomorphic to Pi(F) . Sec. 2.4 Invertibility and Isomorphisms Example 5 Define T : P 3 ( J 2 ) - M 2 x 2 ( i e ) byT( / ) = 103 / ( I ) 1(2) /(3) /(4) It is easily verified that T is linear. By use of the Lagrange interpolation formula in Section 1.6, it can be shown (compare with Exercise 22) that T(/) = O only when / is the zero polynomial. Thus T is one-to-one (see Exercise 11). Moreover, because dim(P3(/?)) = dim(M2x2(i?)), it follows that T is invertible by Theorem 2.5 (p. 71). We conclude that Ps{R) is isomorphic to M2x2(JR). In each of Examples 4 and 5, the reader may have observed that isomor-phic vector spaces have equal dimensions. As the next theorem shows, this is no coincidence. Theorem 2.19. Let V and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if dim(V) = dim(W). Proof. Suppose that V is isomorphic to W and that T: V > W is an isomorphism from V to W. By the lemma preceding Theorem 2.18, we have that dim(V) = dim(W). / Now suppose that dim(V) = dim(W), and let 0 {vi, V2, . ,vn} and 7 = {101,102,... ,wn} be bases for V and W, respectively. By Theorem 2.6 (p. 72), there exists T: V W such that T is linear and T(vi) Wi for i 1 ,2 , . . . ,n. Using Theorem 2.2 (p. 68), we have R(T) = span(T(/?)) = span(7) = W. So T is onto. From Theorem 2.5 (p. 71), we have that T is also one-to-one. Hence T is an isomorphism. 1 By the lemma to Theorem 2.18, if V and W are isomorphic, then either both of V and W are finite-dimensional or both arc infinite-dimensional. Corollary. Let V be a vector space over F. Then V is isomorphic to Fn if and only if dim(V) = n. Up to this point, we have associated linear transformations with their matrix representations. We are now in a position to prove that, as a vector space, the collection of all linear transformations between two given vector spaces may be identified with the appropriate vector space o fmxn matrices. Theorem 2.20. Let V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let 0 and 7 be ordered bases for V and W, respectively. Then the function : (V, W) M m x n ( F ) , defined by $(T) = [T]J for T (V, W), is an isomorphism. L 104 Chap. 2 Linear Transformations and Matrices Proof. By Theorem 2.8 (p. 82), $ is linear. Hence we must show that $ is one-to-one and onto. This is accomplished if we show that for every mxn matrix A, there exists a unique linear transformation T: V W such that $(T) = A. Let 0 {vi,V2,... ,vn}, 7 = {101,103,... ,wm}, and let A be a given mxn matrix. By Theorem 2.6 (p. 72), there exists a unique linear transformation T: V * W such that J(vj) = 5Z AiJWi for 1 - i' - n-But this means that [T]l = A, or $(T) = A. Thus is an isomorphism. 1 Corollary. Let V and W be finite-dimensional vector spaces of dimensions n and rn, respectively. Then (V, W) is finite-dimensional of dimension mn. Proof. The proof follows from Theorems 2.20 and 2.19 and the fact that dim(Mm X n(F)) = mn. I We conclude this section with a result that allows us to see more clearly the relationship between linear transformations defined on abstract finite-dimensional vector spaces and linear transformations from Fn to Fm. We begin by naming the transformation x * [x]p introduced in Sec-tion 2.2. / Definition. Let 0 be an ordered basis for an n-dhncnsional vector space V over the fleid F . The standard representation ofV with respect to 0 is the function Fn defined by p{x) = [x]ff for each x V. Example 6 Let 0 = {(1,0), (0,1)} and 7 = {(1,2), (3,4)}. It is easily observed that 0 and 7 are ordered bases for R2. For x = (1,-2) , we have p (x) = [x](3 f _ 2 j and (j)1(x) = [x - 5 2 We observed earlier that p is a linear transformation. The next theorem tells us much more. Theorem 2.21. For any finite-dimensional vector space V with ordered basis 0, 4>p is an isomorphism. Proof Exercise. 1 This theorem provides us with an alternate proof that an n-dimensional vector space is isomorphic to Fn (sec the corollary to Theorem 2.19). Sec. 2.4 Invertibility and Isomorphisms 105 V (1) -*- w u (2) T -*- F 07 Figure 2.2 Let V and W be vector spaces of dimension n and m, respectively, and let T: V > W be a linear transformation. Define A = [T]l, where 0 and 7 are arbitrary ordered bases of V and W, respectively. We are now able to use 4>@ and 0 7 to study the relationship between the linear transformations T and LA: Fn -> Fm. Let us first consider Figure 2.2. Notice that there are two composites of linear transformations that map V into Fm: 1. Map V into Fn with 0-2. Map V into W with T and follow it by 0 7 to obtain the composite 07T. These two composites are depicted by the dashed arrows in the diagram. By a simple reformulation of Theorem 2.14 (p. 91), we may conclude that LA4>P = 7T; that is, the diagram 'commutes.' Heuristically, this relationship indicates that after V and W are identified with F' and Fm via 0/? and 7, respectively, we may 'identify' T with LA- This diagram allows us to transfer operations on abstract vector spaces to ones on Fn and Fm. Example 7 Recall the linear transformation T: P-.](R) > P2{R) defined in Example 4 of Section 2.2 (T(f(x)) = f'(x)). Let 0 and 7 be the standard ordered bases for P3(R) and P2(i*), respectively, and let (f>p: P-S{R) -> R4 and 0 7 : P2(R) -+ R3 be the corresponding standard representations of P-.i(R) and P2(/^)- If A = [T]}, then / o 1 0 0 A = I 0 0 2 0 ] . 0 0 0 3 / 106 Chap. 2 Linear Transformations and Matrices Consider the polynomial p(x) = 2+x3x2+bx3. We show that Lj(f)p(p(x)) = 07T(p(x)). Now /0 1 0 0s LA4>0(p(x)) = 0 0 2 0 0 0 0 3 / 2 1 - 3 V V But since T(p(x)) = v'(x) = 1 6x + 15a;2, we have -J(vU)) ( - 6 1 15 So LAMPW) = ^ T ( P ( * ) ) . Try repeating Example 7 with different polynomials p(x). EXERCISES 1. Label the following statements as true or false. In each part, V and W are vector spaces with ordered (finite) bases a and 0, respectively, T: V W is linear, and A and B are matrices. (a) ([Tig)'1 [T-'lg. (b) T is invertible if and only if T is one-to-one and onto. (c) T = LA, where A = [T]g. (d) M2X3(F) is isomorphic to F5. (e) Pn{F) is isomorphic to P m (F) if and only if n rn. (f ) AB = I implies that A and B are invertible. (g) If A is invertible, then ( A - 1 ) - 1 = A. (h) A is invertible if and only if L^ is invertible. (i) A must be square in order to possess an inverse. 2. For each of the following linear transformations T, determine whether T is invertible and justify your answer. (a) T: R2 > R3 defined by T(oi,a2) = (ai 2a2 ,a2 ,3ai +4a 2 ) . (b) T: R2 > R3 defined by T(ai ,a 2 ) = (3ai - a 2 ,a 2 ,4ai ) . (c) T: R3 > R3 defined by T(oi,#2,03) = (3ai 2a3,a2,3i +4a 2 ) . (d) T: P3(R) -> P2{R) defined by T(p{x)) = p'(x). (e) T: M2x2(i?) -> P2(R) defined by T (a = a + 26.x + (c + d)x2. (f) T: M2x2(R) -+ M2x2{R) defined by T a + b c a c + d Sec. 2.4 Invertibility and Isomorphisms 107 3. Which of the following pairs of vector spaces are isomorphic? Justify your answers. (a) F3 and P3(F). (b) F 4 a n d P 3 ( F ) . (c) M2x2(i?) and P3(i2). (d) V = {A e M2x2(R): tr(A) = 0} and R4. 4J Let A and B be n x n invertible matrices. Prove that AB is invertible and (AB)'1 = B^A'1. 5.* Let A be invertible. Prove that A1 is invertible and (A1)'1 = (A'1)1. 6. Prove that if A is invertible and AB = O. then B = O. 7. Let A be an n x n matrix. (a) Suppose that A2 = O. Prove that A is not invertible. (b) Suppose that AB O for some nonzero n x n matrix B. Could A be invertible? Explain. 8. Prove Corollaries 1 and 2 of Theorem 2.18. 9. Let A and B be n x n matrices such that AB is invertible. Prove that A and B are invertible. Give an example to show that arbitrary matrices A and B need not be invertible if AB is invertible. 10.T Let A and B be n x n matrices such that AB In. (a) Use Exercise 9 to conclude that A and B are invertible. (b) Prove A B~x (and hence B = A~l). (We are, in effect, saying that for square matrices, a 'one-sided' inverse is a 'two-sided' inverse.) (c) State and prove analogous results for linear transformations de-fined on finite-dimensional vector spaces. 11. Verify that the transformation in Example 5 is one-to-one. 12. Prove Theorem 2.21. 13. Let ~ mean 'is isomorphic to.' Prove that ~ is an equivalence relation on the class of vector spaces over F. 14. Let Construct an isomorphism from V to F . 108 Chap. 2 Linear Transformations and Matrices 15. Let V and W be n-dimensional vector spaces, and let T: V * W be a linear transformation. Suppose that 0 is a basis for V. Prove that T is an isomorphism if and only if T(0) is a basis for W. 16. Let B be an n x n invertible matrix. Define $ : M n x n ( F ) > M n x n (F ) by $(A) = B~XAB. Prove that $ is an isomorphism. 17.* Let V and W be finite-dimensional vector spaces and T: V > W be an isomorphism. Let Vn be a subspace of V. (a) Prove that T(V0) is a subspace of W. (b) Prove that dim(V0) = dim(T(V0)). 18. Repeat Example 7 with the polynomial p(x) = 1 + x + 2x2 + x3. 19. In Example 5 of Section 2.1, the mapping T: M2x2(i?) > M2x2(i?) de-fined by T(M) = Ml for each M e M2x2(i?) is a linear transformation. Let 0 = {EU,E12, E21,E22}, which is a basis for M2x2(R), as noted in Example 3 of Section 1.6. (a) Compute [Tj/3. (b) Verify that LAMM) = ^ T ( M ) for A = [T]^ and / M = 1 2 3 4 20. * Let T: V > W be a linear transformation from an n-dimensional vector space V to an m-dimensional vector space W. Let 0 and 7 be ordered bases for V and W, respectively. Prove that rank(T) = rank(Lyi) and that nullity(T) = nullity (L,i), where A = [T]^. Hint: Apply Exercise 17 to Figure 2.2. 21. Let V and W be finite-dimensional vector spaces with ordered bases 0 = {vi> v2, - - -, Vn} and 7 = {w, w2, - - -, wm}, respectively. By The-orem 2.6 (p. 72), there exist linear transformations T,j: V > W such that Tij(vk) = Wi if k = j 0 iik^ j. First prove that {Tij: 1 < i < m, 1 < j < n} is a basis for (V, W). Then let MlJ be the mxn matrix with 1 in the ith row and j th column and 0 elsewhere, and prove that [T^-H = MlK Again by Theorem 2.6, there exists a linear transformation $ : (V, W) > M m X n (F) such that $(Tij) = My ' . Prove that $ is an isomorphism. Sec. 2.4 Invertibility and Isomorphisms 109 22. Let cn ,c i , . . . , c n be distinct scalars from an infinite field F. Define T: P n (F) - Fn+1 by T( / ) = ( / ( c b ) , / ( d ) , . . . , / (c)). Prove that T is an isomorphism. Hint: Use the Lagrange polynomials associated with C0,Ci,...,C. 23. Let V denote the vector space defined in Example 5 of Section 1.2, and let W = P(F). Define T: V W by T(110 Chap. 2 Linear Transformations and Matrices of vectors in 5. (See Exercise 14 of Section 1.3.) Let # : C{S,F) -> V be defined by * ( / ) = 0 if / is the zero function, and * ( / ) = E / * . sS,/(s)^0 otherwise. Prove that I> is an isomorphism. Thus every nonzero vector space can be viewed as a space of functions. 2.5 THE CHANGE OF COORDINATE MATRIX In many areas of mathematics, a change of variable is used to simplify the appearance of an expression. For example, in calculus an antiderivative of 2xex can be found by making the change of variable u x . The resulting expression is of such a simple form that an antiderivative is easily recognized: 2xex dx = eudu = eu + c = ex + c. Similarly, in geometry the change of variable 2 , 1 , / '= v T 7 f ' can be used to transform the equation 2x2 4xy + by2 1 into the simpler equation (x')2 +6(y')2 = 1, in which form it is easily seen to be the equation of an ellipse. (See Figure 2.4.) We see how this change of variable is determined in Section 6.5. Geometrically, the change of variable is a change in the way that the position of a point P in the plane is described. This is done by introducing a new frame of reference, an x'y'-coordinate system with coordinate axes rotated from the original xy-coordinate axes. In this case, the new coordinate axes are chosen to lie in the direction of the axes of the ellipse. The unit vectors along the x'-axis and the y'-axis form an ordered basis ' { A C 1 _ / - 1 y/E 2 for R2, and the change of variable is actually a change from [P]p = ( j , the coordinate vector of P relative to the standard ordered basis 0 = {ei, e2}, to Sec. 2.5 The Change of Coordinate Matrix 111 [P]pi = ( , J, the coordinate vector of P relative to the new rotated basis 0'. Figure 2.4 A natural question arises: How can a coordinate vector relative to one ba-sis be changed into a coordinate vector relative to the other? Notice that the system of equations relating the new and old coordinates can be represented by the matrix equation 1_ (2 - 1 fx' A u 2 y Notice also that the matrix W y/EV 2 equals [l],, where I denotes the identity transformation on R2. Thus [v]p = Q[v)p> for all v e R2. A similar result is true in general. Theorem 2.22. Let 0 and 0' be two ordered bases for a finite-dimensional vector space V, and let Q [v]%. Then (a) Q is invertible. (b) For any vJ, [v]p - Q[v]p>. Proof, (a) Since lv is invertible, Q is invertible by Theorem 2.18 (p. 101). (b) For any v 6 V, [vb = [W(v)]0 = {vfp,[v]0>=Q[vh> by Theorem 2.14 (p. 91). 112 Chap. 2 Linear Transformations and Matrices The matrix Q = [y]p, defined in Theorem 2.22 is called a change of coor-dinate matrix. Because of part (b) of the theorem, we say that Q changes /^'-coordinates into /^-coordinates. Observe that if 0 {xt,x2,... , xn} and 0' = {x'i,x2, ,x'n}, then X j / WijXi i=l for j = 1 ,2 , . . . , n; that is, the jth column of Q is [x'Ap. Notice that if Q changes /'^'-coordinates into /^-coordinates, then Q~l changes /^-coordinates into /'/-coordinates. (See Exercise 11.) Example 1 In R2, let 0 = {(1,1), (1, -1 )} and 0' = {(2,4), (3,1)}. Since (2,4) = 3(1,1) - 1(1, -1 ) and (3,1) = 2(1,1) + 1(1, -1) , the matrix that changes /^'-coordinates into /^-coordinates is 3 2N Thus, for instance, 1 / 3 [(2A)Yp = Q[(2,4)]p,=Q 0/ V - l For the remainder of this section, we consider only linear transformations that map a vector space V into itself. Such a linear transformation is called a linear operator on V. Suppose now that T is a linear operator on a finite-dimensional vector space V and that 0 and 0' are ordered bases for V. Then V can be represented by the matrices [T]^ and T]p>. What is the relationship between these matrices? The next theorem provides a simple answer using a change of coordinate matrix. Theorem 2.23. Let T be a linear operator on a finite-dimensional vector space V, and let 0 and 0' be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes 0'-coordinates into 0-coordinates. Then Proof. Let I be the identity transformation on V. Then T = IT = Tl; hence, by Theorem 2.11 (p. 88), Q[rp = [']m = [IT]J - [Tl] = [Tfp[fp, = [T]pQ-Therefore [T]p, = Q-^pQ. I Sec. 2.5 The Change of Coordinate Matrix Example 2 Let T be the linear operator on R2 defined by T , (3a b 113 a + 36 7 ' and let 0 and 0' be the ordered bases in Example 1. The reader should verify that [T]a = 3 1 - 1 37' In Example 1, we saw that the change of coordinate matrix that changes /^'-coordinates into /^-coordinates is Q = 3 2 - 1 17 ' and it is easily verified that W 5 1 37 ' Hence, by Theorem 2.23, / [ T ] / 3 ' = Q - 1 [ T ] / 3 Q = ( J I)' To show that this is the correct matrix, we can verify that the image under T of each vector of 0' is the linear combination of the vectors of 0' with the entries of the corresponding column as its coefficients. For example, the image of the second vector in 0' is T 3 - i f i H f i Notice that the coefficients of the linear combination are the entries of the second column of [T]^. It is often useful to apply Theorem 2.23 to compute [T]^, as the next example shows. Example 3 Recall the reflection about the rc-axis in Example 3 of Section 2.1. The rule (x, y) > (x, y) is easy to obtain. We now derive the less obvious rule for the reflection T about the line y = 2x. (See Figure 2.5.) We wish to find an expression for T(a, b) for any (a, b) in R2. Since T is linear, it is completely 114 Chap. 2 Linear Transformations and Matrices y y = 2x Figure 2.5 determined by its values on a basis for R2. Clearly, T(l ,2) = (1,2) and T( -2 ,1) = - ( - 2 , 1 ) = (2, -1 ) . Therefore if we let 0' = 1 1-2 2 7 ' V i then 0' is an ordered basis for R2 and / H > = o - 1 7 ' Let 0 be the standard ordered basis for R2, and let Q be the matrix that changes /^'-coordinates into /^-coordinates. Then 1 - 2 Q = and Q l[^]pQ [T]/9'. We can solve this equation for [T]^ to obtain that [T]li = QTd'Q~l- Because Q 5 I - 2 1 the reader can verify that t * - H i a ) -Since 0 is the standard ordered basis, it follows that T is left-multiplication by [T]p. Thus for any (a,b) in R2, we have T 1 / - 3 4 4 37 b 1 / - 3 a + 46 4 a + 367 ' Sec. 2.5 The Change of Coordinate Matrix 115 A useful special case of Theorem 2.23 is contained in the next corollary, whose proof is left as an exercise. Corollary. Let A e Mn x n(F), and let 7 be an ordered basis for F'. Then [Lyt]7 = Q - 1 AQ, whore Q is the n x n matrix whose jth column is the jth vector of 7. Example 4 Let A = and let 7 = which is an ordered basis for R3. Let Q be the 3 x 3 matrix whose jth column is the j th vector of 7. Then and Q- 1 = So by the preceding corollary, M v : = Q-lAQ = I M V 0 2 4 - 1 8 6 - 1 The relationship between the matrices [T]^' and [T]p in Theorem 2.23 will be the subject of further study in Chapters 5, 6, and 7. At this time, however, we introduce the name for this relationship. Definition. Let A and B be matrices in M n x n (F ) . We say that B is similar to A if there exists an invertible matrix Q such that B = Q.AQ. Observe that the relation of similarity is an equivalence relation (see Ex-ercise 9). So we need only say that A and B are similar. Notice also that in this terminology Theorem 2.23 can be stated as follows: If T is a linear operator on a finite-dimensional vector space V, and if 0 and 0' are any ordered bases for V, then [T]p' is similar to [T]p. Theorem 2.23 can be generalized to allow T: V * W, where V is distinct from W. In this case, we can change bases in V as well as in W (see Exercise 8). 116 Chap. 2 Linear Transformations and Matrices EXERCISES 1. Label the following statements as true or false. (a) Suppose that 0 = {x,x2,...,xn} and 0' = {x i ,x 2 , . . . , xJJ are ordered bases for a vector space and Q is the change of coordinate matrix that changes /^'-coordinates into /^-coordinates. Then the jth column of Q is [xj]p>. Every change of coordinate matrix is invertible. Let T be a linear operator on a finite-dimensional vector space V, let 0 and 0' be ordered bases for V, and let Q be the change of coordinate matrix that changes /^'-coordinates into /^-coordinates. Then [J]p, Q P V Q ' 1 . The matrices A,B M n X n (F) are called similar if B = Q*AQ for some Q Sec. 2.5 The Change of Coordinate Matrix let 0 be the standard ordered basis for R2, and let 0' = Use Theorem 2.23 and the fact that i r - 1 1 2 2 - 1 - 1 1 to find [T]p'. 117 5. Let T be the linear operator on Pi(R) defined by T(p(x)) = p'(x), the derivative of p(x). Let 0 {l,a:} and 0' = {1 + x, 1 x}. Use Theorem 2.23 and the fact that 1 1 1 - i j -1 ( l u 1 2 1 2 6. to find [T]p>. For each matrix A and ordered basis /?, find [L^]^. Also, find an invert-ible matrix Q such that [LA]P = Q~lAQ. (d) A = 7. In R2, let L be the line y = mx, where m ^ 0. Find an expression for T(x, t/), where (a) T is the reflection of R2 about L. (b) T is the projection on L along the line perpendicular to L. (See the definition of projection in the exercises of Section 2.1.) 8. Prove the following generalization of Theorem 2.23. Let T: V W be a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W. Let 0 and 0' be ordered bases for 118 Chap. 2 Linear Transformations and Matrices V, and let 7 and 7' be ordered bases for W. Then [T]j/, = P_ 1[T]JQ, where Q is the matrix that changes /^'-coordinates into /^-coordinates and P is the matrix that changes 7'-coordinates into 7-coordinates. 9. Prove that 'is similar to' is an equivalence relation on MnXn(F). 10. Prove that if A and B are similar n x n matrices, then tr(A) = tr(I?). Hint: Use Exercise 13 of Section 2.3. 11. Let V be a finite-dimensional vector space with ordered bases a,0, and 7. (a) Prove that if Q and R are the change of coordinate matrices that change a-coordinates into ^-coordinates and ,0-coordinates into 7-coordinates, respectively, then RQ is the change of coordinate matrix that changes o>coordinates into 7-coordinates. (b) Prove that if Q changes a-coordinates into /3-coordinates, then Q~l changes /^-coordinates into a-coordinates. 12. Prove the corollary to Theorem 2.23. 13.* Let V be a finite-dimensional vector space over a field F , and let 0 {x, x 2 , . . . , x n } be an ordered basis for V. Let Q be an nxn invertible matrix with entries from F. Define x'j = 22 Qijxi f o r 1 < J < , and set 0' {x[, x2, , x'n}. Prove that 0' is a basis for V and hence that Q is the change of coordinate matrix changing /^'-coordinates into /^-coordinates. 14. Prove the converse of Exercise 8: If A and B are each mxn matrices with entries from a field F , and if there exist invertible mxrn and n x n matrices P and Q, respectively, such that B = P~1AQ, then there exist an n-dimensional vector space V and an m-dimensional vector space W (both over F), ordered bases 0 and 0' for V and 7 and 7' for W, and a linear transformation T: V > W such that A = [T]} and B = [!]},. Hints: Let V = Fn , W = Fm , T = L^, and 0 and 7 be the standard ordered bases for F' and Fm , respectively. Now apply the results of Exercise 13 to obtain ordered bases 0' and 7' from 0 and 7 via Q and P , respectively. Sec. 2.6 Dual Spaces 2.6* DUAL SPACES 119 In this section, we are concerned exclusively with linear transformations from a vector space V into its field of scalars F , which is itself a vector space of di-mension 1 over F . Such a linear transformation is called a linear functional on V. We generally use the letters f, g, h , . . . to denote linear functionals. As we see in Example 1, the definite integral provides us with one of the most important examples of a linear functional in mathematics. Example 1 Let V be the vector space of continuous real-valued functions on the interval [0,27r]. Fix a function g V. The function h: V R defined by 2TT 1 rn h(x) = y x(t)g(t)dt is a linear functional on V. In the cases that g(t) equals sin nt or cos nt, h(x) is often called the n th Fourier coefficient of x. Example 2 Let V = MnXn(F), and define f: V -> F by f(A) = tr(A), the trace of A. By Exercise 6 of Section 1.3, we have that f is a linear functional. Example 3 Let V be a finite-dimensional vector space, and let 0 = {xi,X2,... , x n } be an ordered basis for V. For each i = 1 ,2 , . . . , n, define fi(x) = a, where / i 02 N/3 = dn) is the coordinate vector of x relative to 0. Then fj is a linear functional on V called the i th coordinate function with respect to the basis 0. Note that U(XJ) = Sij, where &y is the Kronecker delta. These linear functionals play an important role in the theory of dual spaces (see Theorem 2.24). Definition. For a vector space V over F, we define the dual space of V to be the vector space (V, F), denoted by V*. Thus V* is the vector space consisting of all linear functionals on V with the operations of addition and scalar multiplication as defined in Section 2.2. Note that if V is finite-dimensional, then by the corollary to Theorem 2.20 (p. 104) dim(V*) = dim((V,F)) = dim(V) dim(F) = dim(V). 120 Chap. 2 Linear Transformations and Matrices Hence by Theorem 2.19 (p. 103), V and V* are isomorphic. We also define the double dual V** of V to be the dual of V*. In Theorem 2.26, we show, in fact, that there is a natural identification of V and V** in the case that V is finite-dimensional. Theorem 2.24. Suppose that V is a finite-dimensional vector space with the ordered basis 0 {x,x2,... ,xn}. Let f* (1 < i < n) be the ith coordi-nate function with respect to 0 as just defined, and let 0* = {fi,f2, ,fn}-Then 0* is an ordered basis for V*, and, for any f G V*, we have f = ^ f ( x i ) f i . i=l Proof Let f V*. Since dim(V*) = n, we need only show that n f = ^ f ( x i ) f i , i= from which it follows that 0* generates V*, and hence is a basis by Corollary 2(a) to the replacement theorem (p. 47). Let g = ^ f ( a : i ) f i . For 1 < j < n, we have g(*j) = f ] [^ f (z i ) f i j (xj) = ^ f ( f f i ) f i ( # j n = f f o ) * y = f ( * i ) . i=l i= Therefore f = g by the corollary to Theorem 2.6 (p. 72). | Definition. Using the notation of Theorem 2.24, we call the ordered basis 0* = {fi,f2, ,fn} of/* that satisfies U(XJ) Sij (1 < i,j < n) the dual basis of 0. Example 4 Let 0 {(2,1), (3,1)} be an ordered basis for R2. Suppose that the dual basis of 0 is given by 0* = {fi, f2}- To explicitly determine a formula for fi, we need to consider the equations 1 = fi(2,1) = fi(2ei + e2) = 2fi(ei) + fi(e2) 0 = f] (3,1) = fi(3ei + e2) = 3fi(ei) + fi(e2). Solving these equations, we obtain fi(ei) = 1 and fi(e2) 3; that is, fi(x, y) x + 3y. Similarly, it can be shown that f2(x,y) = x 2y. Sec. 2.6 Dual Spaces 121 We now assume that V and W are finite-dimensional vector spaces over F with ordered bases 0 and 7, respectively. In Section 2.4, we proved that there is a one-to-one correspondence between linear transformations T: V W and mxn matrices (over F) via the correspondence T *-* [T]2. For a matrix of the form A = [T]l, the question arises as to whether or not there exists a linear transformation U associated with T in some natural way such that U may be represented in some basis as A1. Of course, if m / n, it would be impossible for U to be a linear transformation from V into W. We now answer this question by applying' what we have already learned about dual spaces. Theorem 2.25. Let V and W be finite-dimensional vector spaces over F with ordered bases 0 and 7, respectively. For any linear transformation T: V -> W, the mapping V: W* -> V* defined by T'(g) = gT for all g e W* is a linear transformation with the property that [T']' = ([T]!)'. Proof. For g W*, it is clear that T*(g) = gT is a linear functional on V and hence is in V*. Thus Tf maps W* into V*. We leave the proof that Tb is linear to the reader. To complete the proof, let 0 = {x i ,x 2 , . . . ,x n } and 7 = {3/1,2/2,.. ,ym} with dual bases 0* {fi,f2, ,fn} and 7* = {gi,g2, -/gm}, respectively. For convenience, let A = [T]^. To find the j th column of [Tf]ij,, we be-gin by expressing T (gj) as a linear combination of the vectors of 0*. By Theorem 2.24, we have T t(g J) = g j T = ^(g,T)(x.,)f.s. s=l So the row i, column j entry of [T*]^, is (gjT)(xi) = gj(T(Xi)) = gj I J2 Akiyk j m rn = ^2Akigj(yk) = y~] Akj6jk = Aji. fc=i fc=i Hence [T*] = A*. I The linear transformation Tl defined in Theorem 2.25 is called the trans-pose of T. It is clear that T is the unique linear transformation U such that [ < = cm?)*- _ We illustrate Theorem 2.25 with the next example. 122 Chap. 2 Linear Transformations and Matrices Example 5 Define T: ?i{R) -* R2 by T(p(x)) = (p(0),p(2)). Let 0 and 7 be the standard ordered bases for Pi(R) and R2, respectively. Clearly, m ? - ( 1 j ) We compute [T']r,. directly from the definition. Let /?* = {f 1, f2} ami 7* = {gi,g2}. Suppose that [T*] = P b Then T*(gi) = af, +cf2. So T*(g l)(l) = (ofi + cf2)(l) = afi (1) + cf2(l) = a(l) + c(0) = a. But also (T t(gi))(l) = g i ( T ( l ) ) = g 1 ( l , l ) = l. So a = 1. Using similar computations, we obtain that c = 0, b = 1, and d = 2. Hence a direct computation yields [T / as predicted by Theorem 2.25. - f1 1 0 2 = ( m ? ) We now concern ourselves with demonstrating that any finite-dimensional vector space V can be identified in a natural way with its double dual V**. There is, in fact, an isomorphism between V and V** that does not depend on any choice of bases for the two vector spaces. For a vector x G V, we define x: V* * F by x(f) = f(x) for every f G V*. It is easy to verify that x is a linear functional on V*, so x G V**. The correspondence x *-* x allows us to define the desired isomorphism between V and V**. Lemma. Let V be a finite-dimensional vector space, and let x G V. If x(f) = 0 for all f G V*, then x = 0. Proof. Let x ^ 0. We show that there exists f G V* such that x(f) ^ 0. Choose an ordered basis 0 = {xi,X2,... , x n } for V such that xi = x. Let {fi, f 2 , . . . , fn} be the dual basis of 0. Then fi(xx) = 1 ^ 0. Let f = f,. I Theorem 2.26. Let V be a finite-dimensional vector space, and define ip: V V** by ip(x) = x. Then ip is an isomorphism. Sec. 2.6 Dual Spaces Proof, (a) if) is linear: Let x,y G V and c F. For f G V*, we have V>(cx + y)(f) = f(cx + y) = ef(x) + % ) = cx(f) + y(f) = (cx + y)(f). 123 Therefore -0(cx + y) = ex + y = cip(x) + V(y)-(b) {/> is one-to-one: Suppose that i})(x) is the zero functional on V* for some x G V. Then x(f) = 0 for every f G V*. By the previous lemma, we conclude that x = 0. (c) ijj is an isomorphism: This follows from (b) and the fact that dim(V) = dim(V**). | Corollary. Let V be a finite-dimensional vector space with dual space V*. Then every ordered basis for V* is the dual basis for some basis for V. Proof. Let {fi,f2 , . . . ,fn} be an ordered basis for V*. We may combine Theorems 2.24 and 2.26 to conclude that for this basis for V* there exists a dual basis {xi,X2,... , x n } in V**, that is, 5ij = x~i(fj) = f/(xi) for all i and j. Thus {fi, f2,..., fn} is the dual basis of {xi, x2,..., x^ | . I! Although many of the ideas of this section, (e.g., the existence of a dual space), can be extended to the case where V is not finite-dimensional, only a finite-dimensional vector space is isomorphic to its double dual via the map x x. In fact, for infinite-dimensional vector spaces, no two of V, V*, and V** are isomorphic. EXERCISES 1. Label the following statements as true or false. Assume that all vector spaces are finite-dimensional. (a) Every linear transformation is a linear functional. (b) A linear functional defined on a field may be represented as a 1 x 1 matrix. (c) Every vector space is isomorphic to its dual space. (d) Every vector space is the dual of some other vector space. (e) If T is an isomorphism from V onto V* and 0 is a finite ordered basis for V, then T(0) = 0*. (f) If T is a linear transformation from V to W, then the domain of (vy is v**. (g) If V is isomorphic to W, then V* is isomorphic to W*. y2 + z2 124 Chap. 2 Linear Transformations and Matrices (h) The derivative of a function may be considered as a linear func-tional on the vector space of differentiable functions. 2. For the following functions f on a vector space V, determine which are linear functionals. (a) V = P(R); f(p(x)) = 2p'(0) + p'(l), where ' denotes differentiation (b) V = R2;f(x,2/) = (2x,4y) (c) V = M 2 x 2 (F) ; f ( i4)=tr ( i4) (d) V = R 3 ; f (x ,y ,z )=a ; 2 (e) V = P(/l);f(p(x))=J01p(t) R2 by T(x,y) = (Zx + 2y,x). (a) Compute T*(f). (b) Compute [T^p*, where 0 is the standard ordered basis for R2 and 0* = {f 1, f2} is the dual basis, by finding scalars a, b, c, and d such that T'(fi) = ofi + cf2 and T'(f2) = 6f: + df2. (c) Compute [T]p and ([T]^)4, and compare your results with (b). 7. Let V = Pi(R) and W = R2 with respective standard ordered bases 0 and 7. Define T: V > W by T(p(x)) = (p(0) - 2p(l),p(0) +p'(0)), where p'(x) is the derivative of p(x). Sec. 2.6 Dual Spaces 125 8. 9. (a) For f G W* defined by f (o, b) = a - 26, compute T'(f). (b) Compute [T*]^, without appealing to Theorem 2.25. (c) Compute [T] and its transpose, and compare your results with (b). Show that every plane through the origin in R3 may be identified with the null space of a vector in (R3)*. State an analogous result for R2. Prove that a function T: Frt Fm is linear if and only if there exist fi,f2,--.,fm G (Fn)* such that T(x) = (fi(x),f2(x),... ,fm(x)) for all x G Fn. Hint: If T is linear, define fj(x) = (g;T)(x) for x G Fn; that is, fi T'(gi) for 1 < i < m, where {gi,g2, ,gm} is the dual basis of the standard ordered basis for Fm . 10. Let V = Pn(F), and let cn ,c i , . . . , cn be distinct scalars in F . (a) For 0 < i < n, define U G V* by U(p(x)) = p(c.i). Prove that {fo, f i , . . . ,fn} is a basis for V*. Hint: Apply any linear combi-nation of this set that equals the zero transformation to p(x) = (x ci)(x c2) (x cn), and deduce that the first coefficient is zero. (b) Use the corollary to Theorem 2.26 and (a) to show that there exist unique polynomials po(x),pi(x), . . . ,pn(x) such that Pi(cj) = 5ij for 0 < i < n. These polynomials are the Lagrange polynomials defined in Section 1.6. (c) For any scalars an, fli,..., a n (not necessarily distinct), deduce that there exists a unique polynomial q(x) of degree at most n such that q(ci) a,i for 0 < i < n. In fact, a ( x ) = ^2126 Chap. 2 Linear Transformations and Matrices Suppose now that i(b a) . _ i Ci = a- for % = 0 ,1 , . . . , n. n For n = 1, the preceding result yields the trapezoidal rule for evaluating the definite integral of a polynomial. For n 2, this result yields Simpson's rule for evaluating the definite integral of a polynomial. 11. Let V and W be finite-dimensional vector spaces over F , and let tp and ip2 be the isomorphisms between V and V** and W and W**, respec-tively, as defined in Theorem 2.26. Let T: V W be linear, and define T ' = (T*)*. Prove that the diagram depicted in Figure 2.6 commutes (i.e., prove that ip2T = T ' ^ ) . V V W i>2 W / Figure 2.6 12. Let V be a finite-dimensional vector space with the ordered basis 0. Prove that ip(0) = 0**, where ip is defined in Theorem 2.26. In Exercises 13 through 17, V denotes a finite-dimensional vector space over F . For every subset S of V, define the annihi la tor 5 of 5 as 5 = {f G V*: f(x) = 0 for all x G S}. 13. (a) Prove that S is a subspace of V*. (b) If W is a subspace of V and x g' W, prove that there exists f G W such that f (x) ^ 0. (c) Prove that (5) = span(^'(S')), where 'ip is defined as in Theo-rem 2.26. (d) For subspaces Wi and W2, prove that Wi = W2 if and only if W? = Wg. (e) For subspaces Wi and W2, show that (Wj + W2) = Wj n W. 14. Prove that if W is a subspace of V, then dim(W) + dim(W) = dim(V). Hint: Extend an ordered basis {xi,X2, , xk] of W to an ordered ba-sis 0 = {xi,X2,. -. , x n } of V. Let 0* = {fi,f2,... ,fn}. Prove that {ffc+i,ffc+2,...,f} is a basis for W. Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 127 15. Suppose that W is a finite-dimensional vector space and that T: V W is linear. Prove that N(T 0, let F(t) denote the force acting on the weight and y(t) denote the position of the weight along the y-axis. For example, y(0) = s. The second derivative of y with respect 128 Chap. 2 Linear Transformations and Matrices Figure 2.7 to time, y'(t), is the acceleration of the weight at time t hence, by Newton's second law of motion, F(t) = my'(t). (1) It is reasonable to assume that the force acting on the weight is due totally to the tension of the spring, and that this force satisfies Hooke's law: The force acting on the weight is proportional to its displacement from the equilibrium position, but acts in the opposite direction. If k > 0 is the proportionality constant, then Hooke's law states that F(t) = -ky(t). Combining (1) and (2), we obtain my' = ky or (2) y + y = 0. (3) The expression (3) is an example of a differential equation. A differential equation in an unknown function y = y(t) is an equation involving y, t, and derivatives of y. If the differential equation is of the form any (n) an-iy (n-1) + ay (i) a>oy = f; (4) where an, a i , . . . , on and / are functions of t and y^k' denotes the kth deriva-tive of y, then the equation is said to be linear. The functions ai are called the coefficients of the differential equation (4). Thus (3) is an example of a linear differential equation in which the coefficients are constants and the function / is identically zero. When / is identically zero, (4) is called homogeneous. In this section, we apply the linear algebra we have studied to solve ho-mogeneous linear differential equations with constant coefficients. If an ^ 0, Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 129 we say that differential equation (4) is of order n. In this case, we divide both sides by an to obtain a new, but equivalent, equation .'/ (n) K-iy (n-1) + + hy{1) + b0y = 0, where 6, = a ; / a n for i = 0 ,1 , . . . , n 1. Because of this observation, we always assume that the coefficient an in (4) is 1. A solution to (4) is a function that when substituted for y reduces (4) to an identity. Example 1 The function y(t) = sin y/k/m t is a solution to (3) since y (t) + y(t) sin t + sin m m V m m m for all f. Notice, however, that substituting y(t) = t into (3) yields k k y'(t) + -y(t) = -t, m m which is not identically zero. Thus y(t) t is not a solution to (3). In our study of differential equations, it is useful to regard solutions as complex-valued functions of a real variable even though the solutions that are meaningful to us in a physical sense are real-valued. The convenience of this viewpoint will become clear later. Thus we are concerned with the vector space J-~(R, C) (as defined in Example 3 of Section 1.2). In order to consider complex-valued functions of a real variable as solutions to differential equations, we must define what it means to differentiate such functions. Given a complex-valued function x G T(R, C) of a real variable t, there exist unique real-valued functions xi and X2 of i, such that x(t) = xi(t) + ix2(t) for t G R, where i is the imaginary number such that i2 = 1. We call Xi the real part and X2 the imaginary part of x. Definitions. Given a function x G J-(R, C) with real part x and imag-inary part x2, we say that x is differentiable ifx and x2 are differentiable. If x is differentiable, we define the derivative x' of x by 3s 'dj i ~~ IXo' We illustrate some computations with complex-valued functions in the following example. 130 Chap. 2 Linear Transformations and Matrices Example 2 Suppose that x(t) cos 2t + i sin 2t. Then x'(t) = -2 sin 2t + 2i cos 2t. We next find the real and imaginary parts of x2. Since x2(t) = (cos 2t + i sin 2t)2 = (cos2 2t - sin2 2t) + i(2 sin 2t cos 2i) = cos4i + isin4t, the real part of x2(t) is cos4i, and the imaginary part is sin4. The next theorem indicates that we may limit our investigations to a vector space considerably smaller than T(R, C). Its proof, which is illustrated in Example 3, involves a simple induction argument, which we omit. Theorem 2.27. Any solution to a homogeneous linear differential equa-tion with constant coefficients has derivatives of all orders; that is, if x is a solution to such an equation, then x^k' exists for every positive integer k. Example 3 To illustrate Theorem 2.27, consider the equation / y(2) + 4y = 0. Clearly, to qualify as a solution, a function y must have two derivatives. If y is a solution, however, then y(2) = -4y-Thus since y^ is a constant multiple of a function y that has two derivatives, yW must have two derivatives. Hence y^ exists; in fact, 3,(4) = -4yM. Since j / 4 ' is a constant multiple of a function that we have shown has at least two derivatives, it also has at least two derivatives; hence y(6> exists. Continuing in this manner, we can show that any solution has derivatives of all orders. Definition. We use C to denote the set of all functions in T(R, C) that have derivatives of all orders. It is a simple exercise to show that C is a subspace of F(R, C) and hence a vector space over C. In view of Theorem 2.27, it is this vector space that Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 131 is of interest to us. For x G C, the derivative x' of x also lies in C. We can use the derivative operation to define a mapping D: C + C by D(x) = x' forxGC 0 0 . It is easy to show that D is a linear operator. More generally, consider any polynomial over C of the form p(t) = antn +an-it n- l + at + OQ. If we define p(D) = anDn + On-iD*-1 + + o iD + o0l, then p(D) is a linear operator on C. (See Appendix E.) Definitions. For any polynomial p(t) over C of positive degree, p(D) is called a differential operator. The order of the differential operator p(D) is the degree of the polynomial p(t). Differential operators are useful since they provide us with a means of reformulating a differential equation in the context of linear algebra. Any homogeneous linear differential equation with constant coefficients, y{n)+an-iy(n-v + --- + aiy{i) +a0y=(), can be rewritten using differential operators as ;Dn + o w _iD n - 1 + aiD + ool)(y) = 0. Definition. Given the differential equation above, the complex polyno-mial p(t) = tn + o n _ i i n _ 1 + + oii + a0 is called the auxiliary polynomial associated with the equation. For example, (3) has the auxiliary polynomial Any homogeneous linear differential equation with constant coefficients can be rewritten as p(D)(y) = 0, where p(t) is the auxiliary polynomial associated with the equation. Clearly, this equation implies the following theorem. 132 Chap. 2 Linear Transformations and Matrices Theorem 2.28. The set of all solutions to a homogeneous linear differen-tial equation with constant coefficients coincides with the null space ofp(D), where p(t) is the auxiliary polynomial associated with the equation. Proof Exercise. I Corollary. The set of all solutions to a homogeneous linear differential equation with constant coefficients is a subspace of C. In view of the preceding corollary, we call the set of solutions to a homo-geneous linear differential equation with constant coefficients the solution space of the equation. A practical way of describing such a space is in terms of a basis. We now examine a certain class of functions that is of use in finding bases for these solution spaces. For a real number s, we are familiar with the real number e9, where e is the unique number whose natural logarithm is 1 (i.e., lne = 1). We know, for instance, certain properties of exponentiation, namely, es+t = eset and e -r for any real numbers s and t. We now extend the definition of powers of e to include complex numbers in such a way that these properties are preserved. Definition. Let 6 = a + ib be a complex number with real part a and imaginary part b. Define ec = ea(cosb + isinb). The special case Jb e = cos b + i sin 6 is called Euler's formula. For example, for c = 2 + i(n/Z), = e ( c o s - + J s i n - ) = e ^ -Clearly, if c is real (6 = 0), then we obtain the usual result: ec = ea . Using the approach of Example 2, we can show by the use of trigonometric identities that ec+d = eced and e'c = for any complex numbers c and d. Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 133 Definition. A function f:R*C defined by f(t) = ect for a fixed complex number c is called an exponential function. The derivative of an exponential function, as described in the next theo-rem, is consistent with the real version. The proof involves a straightforward computation, which we leave as an exercise. Theorem 2.29. For any exponential function f(t) = ect, f'(t) = cect. Proof. Exercise. 1 We can use exponential functions to describe all solutions to a homoge-neous linear differential equation of order 1. Recall that the order of such an equation is the degree of its auxiliary polynomial. Thus an equation of order 1 is of the form y' + a0y = 0. (5) Theorem 2.30. The solution space for (5) is of dimension 1 and has {e~aot} as a basis. Proof. Clearly (5) has e _ a , | t as a solution. Suppose that x(t) is any solution to (5). Then x'(t) = -a0x(t) for all t G R. y Define z(t) = eai,tx(t). Differentiating z yields z'(t) = (eaot)'x(t) + eV(t) = o0e*x(*) - a0eaotx(t) = 0. (Notice that the familiar product rule for differentiation holds for complex-valued functions of a real variable. A justification of this involves a lengthy, although direct, computation.) Since z' is identically zero, z is a constant function. (Again, this fact, well known for real-valued functions, is also true for complex-valued functions. The proof, which relies on the real case, involves looking separately at the real and imaginary parts of z.) Thus there exists a complex number k such that z(t) = eaotx(t) = k for all t G R. So x(t) = ke-aot. We conclude that any solution to (5) is a linear combination of e~aot. 134 Chap. 2 Linear Transformations and Matrices Another way of stating Theorem 2.30 is as follows. Corollary. For any complex number c, the null space of the differential operator D cl has {ect} as a basis. We next concern ourselves with differential equations of order greater than one. Given an nth order homogeneous linear differential equation with constant coefficients, ^ + a n . l 2 / ( - 1 ' + OiyW + a 0 ? / = 0, its auxiliary polynomial p(t) = tn + an_xtn'1 + + ait + oo factors into a product of polynomials of degree 1, that is, p(t) = (t-ci)(t-c2)---(t-cn), where c.i,c2,... ,cn are (not necessarily distinct) complex numbers. (This follows from the fundamental theorem of algebra in Appendix D.) Thus p(D) = (D-c1)(D-c2)---(D-cn). The operators D c j commute, and so, by Exercise 9, we have that r, N(D-Cil) C N(p(D)) foral l i Since N(p(D)) coincides with the solution space of the given differential equa-tion, we can deduce the following result from the preceding corollary. Theorem 2.31. Let p(t) be the auxiliary polynomial for a homogeneous linear differential equation with constant coefficients. For any complex num-ber c, if c is a zero ofp(t), then ect is a solution to the differential equation. Example 4 Given the differential equation y'-3y' + 2y=0, its auxiliary polynomial is p(t) = t2 - U + 2 = (t - l)(t - 2). Hence, by Theorem 2.31, el and e2t are solutions to the differential equa-tion because c 1 and c 2 are zeros of p(t). Since the solution space of the differential equation is a subspace of C, span({e*,e2*}) lies in the solution space. It is a simple matter to show that {ee,e2t} is linearly inde-pendent. Thus if we can show that the solution space is two-dimensional, we can conclude that {e*,e2*} is a basis for the solution space. This result is a consequence of the next theorem. Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 135 Theorem 2.32. For any differential operator p(D) of order n, the null space ofp(D) is an n-dimensional subspace of C. As a preliminary to the proof of Theorem 2.32, we establish two lemmas. Lemma 1. The differential operator D c: C > C is onto for any complex number c. Proof. Let v G C. We wish to find a u G C such that (D c)u = v. Let w(t) = v(t)e~ct for t G R. Clearly, w G C because both v and e~ct lie in C. Let W and w2 be the real and imaginary parts of w. Then w and w2 are continuous because they are differentiable. Hence they have antiderivatives, say, W and W2, respectively. Let W: R, * C be defined by W(t) = Wt (t) + iW2(t) for t G R. Then W G C, and the real and imaginary parts of W are W and W2, respectively. Furthermore, W = w. Finally, let u: R * C be defined by u(t) = W(t)ect for t G R. Clearly u G C, and since (D - c)u(t) = u'(t) - cu(t) = W'(t)cct + W(t)cect - cW(f)ect = w(t)e,:t = v(t)erctert = v(t), we have (D c)u = v. II Lemma 2. Let V be a vector space, and suppose that T and U are linear operators on V such that U is onto and the null spaces of T and U are finite-dimensional. Then the null space of TU is finite-dimensional, and dim(N(TU)) = dim(N(T)) + dim(N(U)). Proof. Let p dim(N(T)), q = dim(N(U)), and {ui,u2,... ,up} and {v,v2,... ,Vq] be bases for N(T) and N(U), respectively. Since U is onto, we can choose for each i (1 < i < p) a vector Wi G V such that U(wi) m. Note that the Wi's are distinct. Furthermore, for any i and j, Wi 7^ Vj, for otherwise Ui = i)(wi) = U(VJ) = 0a contradiction. Hence the set 0= {wi,w2,... ,wp,vi,v2,... ,vq} contains p + q distinct vectors. To complete the proof of the lemma, it suffices to show that 0 is a basis for N(TU). 136 Chap. 2 Linear Transformations and Matrices We first show that 0 generates N(TU). Since for any Wi and Vj in 0, T U K ) = T(ui) = 0 and TU(^) = T(0) = 0, it follows that 0 C N(TU). Now suppose that v G N(TU). Then 0 - TU(v) = T(U(v)). Thus U(v) N(T). So there exist scalars 01,02,.. . ,op such that J(v) = aiu + 02^2 + V apup = aiU(w) + a2U(w2) H h aPU(wp) = J(ttiWi + 02^2 + f- npwp). Hence J(v (aiwi + 02^2 H f- aPwp)) = 0. Consequently, v (aWi + 02^2 + + apwp) lies in N(U). It follows that there exist scalars 61, b2,..., bq such that v - (aiwi + a2w2 H r- apwp) = 61^1 + &2W2 H h &?u9 or v = aiu/i + 02^2 H f- apwp + 61 vi + 62^2 H r- 6gv9. Therefore /? spans N(TU). To prove that 0 is linearly independent, let ai,a2,... ,ap,b,b2,... ,bq be any scalars such that aiwi + a2u'2 H h apwp + &1V1 + 62^2 H r- &9vg = 0. (6) Applying U to both sides of (6), we obtain oiUi + a2u2 + + apup 0. Since {m,u2,... ,up] is linearly independent, the a;'s are all zero. Thus (6) reduces to &if 1 + b2v2 '1 r bqvq = 0. Again, the linear independence of {v,v2,... ,vq} implies that the frj's are all zero. We conclude that 0 is a basis for N(TU). Hence N(TU) is finite-dimensional, and dim(N(TU)) = p + q = dim(N(T)) + dim(N(U)). I Proof of Theorem 2.32. The proof is by mathematical induction on the order of the differential operator p(D). The first-order case coincides with Theorem 2.30. For some integer n > 1, suppose that Theorem 2.32 holds for any differential operator of order less than n, and consider a differential Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 137 operator p(D) of order n. The polynomial p(t) can be factored into a product of two polynomials as follows: p(t) = q(t)(t-c), where q(t) is a polynomial of degree n 1 and c is a complex number. Thus the given differential operator may be rewritten as p(D) = g(D)(D-cl) . Now, by Lemma 1, D cl is onto, and by the corollary to Theorem 2.30, d im(N(D-d)) = 1. Also, by the induction hypothesis, dim(N(r/(D)) = n1. Thus, by Lemma 2, we conclude that dim(N(p(D))) = dim(N(138 Chap. 2 Linear Transformations and Matrices Since the auxiliary polynomial factors as (t + 4)(t + 1), it has two distinct zeros, 1 and 4. Thus {e~L,e~4t} is a basis for the solution space. So any solution to the given equation is of the form y(t)=ble-t + b2e-4t for unique scalars b and b2. Example 6 We find all solutions to the differential equation y' + 9y = 0. The auxiliary polynomial t2 + 9 factors as (t 3i)(t + 3?') and hence has distinct zeros Ci = 3i and c2 3i. Thus {e3*f,e-3**} is a basis for the solution space. Since cos 3^ = - ( e 3/7 + e~3i t) and sin3 = (e :nt -3it 2i ), it follows from Exercise 7 that {cos St, sin 3t} is also a basis for this solution space. This basis has an advantage over the original one because it consists of the familiar sine and cosine functions and makes no reference to the imaginary number i. Using this latter basis, we see that any solution to the given equation is of the form y(t) = b cos St + b2 sin 3t for unique scalars &iand b2. Next consider the differential equation y' + 2y' + y = 0, for which the auxiliary polynomial is (t + l)2 . By Theorem 2.31, e~l is a solution to this equation. By the corollary to Theorem 2.32, its solution space is two-dimensional. In order to obtain a basis for the solution space, we need a solution that is linearly independent of e _ t . The reader can verify that te~l is a such a solution. The following lemma extends this result. Lemma. For a given complex number c and positive integer n, suppose that (t c)n is the auxiliary polynomial of a homogeneous linear differential equation with constant coefficients. Then the set 0={ect,tect,...,tn-1ect} is a basis for the solution space of the equation. Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 139 Proof. Since the solution space is n-dimensional, we need only show that 0 is linearly independent and lies in the solution space. First, observe that for any positive integer k, k- ct Lkct Lkct Hence for k < n, D - c)(tKect) = ktK~lect + ctKect - ctKe = ktk~1ect. (D-c)n(tkect) = 0. It follows that 0 is a subset of the solution space. We next show that 0 is linearly independent. Consider any linear combi-nation of vectors in 0 such that ,Tl-lct b0ect + hie' + + bn-itn-*ea = 0 for some scalars bo, &i,. . . , 6n- i- Dividing by ect in (7), we obtain &n + M n-l bn^tn~1 = 0. (7) (8) y(4) _ 4y(3) + yK Thus the left side of (8) must be the zero polynomial function. We conclude that the coefficients bo,bi,...,bni are all zero. So 0 is linearly independent and hence is a basis for the solution space. f] Example 7 We find all solutions to the differential equation Jy(2) - ^ ( l ) +y=0. Since the auxiliary polynomial is t4 - 4t6 + 6t2 -4t+l = (t- l)4 , we can immediately conclude by the preceding lemma that {e*, tef, fte1:, i3e*} is a basis for the solution space. So any solution y to the given differential equation is of the form y(t) = bie1 + bite* + M2e* + ht3e* for unique scalars b, b2, bj, and 64. The most general situation is stated in the following theorem. Theorem 2.34. Given a homogeneous linear differential equation with constant coefficients and auxiliary polynomial (t-Ci)ni(t-c2)n*-'-(t-ck)n where n,n2,...,nk arc positive integers and c,,c2,... ,ck are distinct com-plex numbers, the following set is a basis for the solution space of the equation: {eClt,teCl ... ,tni-1ec,t,... ,eCfct,ieCfct,... ,tn' eCkt}. 140 Chap. 2 Linear Transformations and Matrices Proof. Exercise. | Example 8 The differential equation y(3) _ 4y(2) + 5^(1) -2y= 0 has the auxiliary polynomial f3 - 42 + 5t - 2 = (* - l)2(t - 2). By Theorem 2.34, {e*,e*,e2Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 141 2. For each of the following parts, determine whether the statement is true or false. Justify your claim with either a proof or a counterexample, whichever is appropriate. (a) Any finite-dimensional subspace of C is the solution space of a homogeneous linear differential equation with constant coefficients. (b) There exists a homogeneous linear differential equation with con-stant coefficients whose solution space has the basis {t, t2}. (c) For any homogeneous linear differential equation with constant coefficients, if a; is a solution to the equation, so is its derivative x'. Given two polynomials p(t) and q(t) in P(C), if x G N(p(D)) and y G N((/(D)), then (d) x + yeU(P(D)q(D)). (e) xyeU(p(D)q(D)). 3. Find a basis for the solution space of each of the following differential equations. (a) y' + 2y' + y = 0 (b) */' = ; + y = 0 (d) y' + 2y, + y=0 (e) y(3) - y{2) + 3y{1) +5y = 0 / 4. Find a basis for each of the following subspaces of C. (a) N ( D 2 - D - I ) (b) N(D 3 -3D 2 + 3 D - I ) (c) N(D3 + 6D2+8D) 5. Show that C is a subspace of T(R, C). 6. (a) Show that D: C * C is a linear operator. (b) Show that any differential operator is a linear operator on C01 7. Prove that if {x, y} is a basis for a vector space over C, then so is {=o + v),^-y) 8. Consider a second-order homogeneous linear differential equation with constant coefficients in which the auxiliary polynomial has distinct con-jugate complex roots a + ib and a ib, where a,b G R. Show that {eat cos bt, eat sin bt} is a basis for the solution space. 142 Chap. 2 Linear Transformations and Matrices 9. Suppose that {Ui, U2, , Un} is a collection of pairwise commutative linear operators on a vector space V (i.e., operators such that U7Uj = UjUi for all i,j). Prove that, for any i (1 < i < n), N(U i)CN(UiU2---Un) . 10. Prove Theorem 2.33 and its corollary. Hint: Suppose that beClt + b2eC2t - h bneCnt = 0 (where the c^'s are distinct). To show the b^s are zero, apply mathematical induction on n as follows. Verify the theorem for n = 1. Assuming that the theorem is true for n 1 functions, apply the operator D cnl to both sides of the given equation to establish the theorem for n distinct exponential functions. 11. Prove Theorem 2.34. Hint: First verify that the alleged basis lies in the solution space. Then verify that this set is linearly independent by mathematical induction on k as follows. The case k = 1 is the lemma to Theorem 2.34. Assuming that the theorem holds for k 1 distinct Cj's, apply the operator (D Cfcl)'fc to any linear combination of the alleged basis that equals 0. 12. Let V be the solution space of an nth-order homogeneous linear differ-ential equation with constant coefficients having auxiliary polynomial p(t). Prove that if p(t) = g(t)h(t), where g(t) and h(t) are polynomials of positive degree, then N(MD)) = R(o(Dv)) = p(D)(V), where Dv: V > V is defined by Dv(x) = x' for x G V. Hint: First prove o(D)(V) C N(/i(D)). Then prove that the two spaces have the same finite dimension. 13. A differential equation y (n ) + an-iy{n-l) + + alV{l) + a0y = x is called a nonhomogeneous linear differential equation with constant coefficients if the a% 's are constant and x is a function that is not iden-tically zero. (a) Prove that for any x G C there exists y G C such that y is a solution to the differential equation. Hint: Use Lemma 1 to Theorem 2.32 to show that for any polynomial pit), the linear operator p(D): C C is onto. Sec. 2.7 Homogeneous Linear Differential Equations with Constant Coefficients 143 (b) Let V be the solution space for the homogeneous linear equation +Oii(/(1) +a0y= 0. y ^ + o n - i ^ ' - 1 ) Prove that if z is any solution to the associated nonhomogeneous linear differential equation, then the set of all solutions to the nonhomogeneous linear differential equation is [z + y: ye V}. 14. Given any nth-order homogeneous linear differential equation with con-stant coefficients, prove that, for any solution x and any to G R, if x(t0) = x'(to) = = .T(TC_1)(0) = 0, then x = 0 (the zero function). Hint: Use mathematical induction on n as follows. First prove the con-clusion for the case n = 1. Next suppose that it is true for equations of order n 1, and consider an nth-order differential equation with aux-iliary polynomial p(t). Factor p(t) = q(t)(t c), and let z = q((D))x. Show that z(to) = 0 a n d z' cz 0 to conclude that z = 0. Now apply the induction hypothesis. 15. Let V be the solution space of an nth-order homogeneous linear dif-ferential equation with constant coefficients. Fix ta G R, and define a mapping $ : V + C by * ( * ) = * x(to) x'(to) xSn-lHto)J for each x in V. (a) (b) Prove that $ is linear and its null space is the zero subspace of V. Deduce that is an isomorphism. Hint: Use Exercise 14. Prove the following: For any nth-order homogeneous linear dif-ferential equation with constant coefficients, any to G R, and any complex numbers Co, Ci, . . . ,c n_i (not necessarily distinct), there exists exactly one solution, x, to the given differential equation such that x(to) co and x^(to) = ck for k = 1, 2 , . . . n 1. 16. Pendular Motion. It is well known that the motion of a pendulum is approximated by the differential equation 0' + ^8= 0, where 6(t) is the angle in radians that the pendulum makes with a vertical line at time t (see Figure 2.8), interpreted so that 6 is positive if the pendulum is to the right and negative if the pendulum is to the 144 Chap. 2 Linear Transformations and Matrices Figure 2.8 left of the vertical line as viewed by the reader. Here I is the length of the pendulum and g is the magnitude of acceleration due to gravity. The variable t and constants / and g must be in compatible units (e.g., t in seconds, / in meters, and g in meters per second per second). (a) Express an arbitrary solution to this equation as a linear combi-nation of two real-valued solutions. (b) Find the unique solution to the equation that satisfies the condi-tions 0(0) = 0O > 0 and 0'(O) = 0. (The significance of these conditions is that at time t = 0 the pendulum is released from a position displaced from the vertical by 0n.) (c) Prove that it takes 2-Ky/lJg units of time for the pendulum to make one circuit back and forth. (This time is called the period of the pendulum.) 17. Periodic Motion of a Spring without Damping. Find the general solu-tion to (3), which describes the periodic motion of a spring, ignoring frictional forces. 18. Periodic Motion of a Spring with Damping. The ideal periodic motion described by solutions to (3) is due to the ignoring of frictional forces. In reality, however, there is a frictional force acting on the motion that is proportional to the speed of motion, but that acts in the opposite direction. The modification of (3) to account for the frictional force, called the damping force, is given by my + ry' + ky 0, where r > 0 is the proportionality constant, (a) Find the general solution to this equation. Chap. 2 Index of Definitions 145 (b) Find the unique solution in (a) that satisfies the initial conditions y(0) 0 and y'(0) = vo, the initial velocity. (c) For y(t) as in (b), show that the amplitude of the oscillation de-creases to zero; that is, prove that lim y(t) = 0. t>oo 19. In our study of differential equations, we have regarded solutions as complex-valued functions even though functions that are useful in de-scribing physical motion are real-valued. Justify this approach. 20. The following parts, which do not involve linear algebra, arc included for the sake of completeness. (a) Prove Theorem 2.27. Hint: Use mathematical induction on the number of derivatives possessed by a solution. (b) For any c,d G C, prove that ec+d = cced and (c) Prove Theorem 2.28. (d) Prove Theorem 2.29. (e) Prove the product rule for differentiating complex-valued func-tions of a real variable: For any differentiable functions x and y in T(R, C), the product xy is differentiable and (xy)' = x'y + xy'. Hint: Apply the rules of differentiation to the real and imaginary parts of xy. (f) Prove that if x G J-(R, C) and x' 0, then x is a constant func-tion. INDEX OF DEFINITIONS FOR CHAPTER 2 Auxiliary polynomial 131 Change of coordinate matrix 112 Clique 94 Coefficients of a differential equation 128 Coordinate function 119 Coordinate vector relative to a basis 80 Differential equation 128 Differential operator 131 Dimension theorem 69 Dominance relation 95 Double dual 120 Dual basis 120 Dual space 119 Euler's formula 132 Exponential function 133 Fourier coefficient 119 Homogeneous linear differential equation 128 Identity matrix 89 Identity transformation 67 146 Chap. 2 Linear Transformations and Matrices Incidence matrix 94 Inverse of a linear transformation 99 Inverse of a matrix 100 Invertible linear transformation 99 Invertible matrix 100 Isomorphic vector spaces 102 Isomorphism 102 Kronecker delta 89 Left-multiplication transformation 92 Linear functional 119 Linear operator 112 Linear transformation 65 Matrix representing a linear trans-formation 80 Nonhomogeneous differential equa-tion 142 Nullity of a linear transformation 69 Null space 67 Ordered basis 79 Order of a differential equation 129 Order of a differential operator 131 Product of matrices 87 Projection on a subspace 76 Projection on the x-axis 66 Range 67 Rank of a linear transformation 69 Reflection about the x-axis 66 Rotation 66 Similar matrices 115 Solution to a differential equation 129 Solution space of a homogeneous dif-ferential equation 132 Standard ordered basis for F' 79 Standard ordered basis for Pn(F) 79 Standard representation of a vector space with respect to a basis 104 Transpose of a linear transformation 121 Zero transformation 67 / 3 E l e m e n t a r y M a t r i x O p e r a t i o n s a n d S y s t e m s o f L i n e a r E q u a t i o n s 3.1 Elementary Matrix Operations and Elementary Matrices 3.2 The Rank of a Matrix and Matrix Inverses 3.3 Systems of Linear EquationsTheoretical Aspects 3.4 Systems of Linear EquationsComputational Aspects _. his chapter is devoted to two related objectives: 1. the study of certain 'rank-preserving' operations on matrices; 2. the application of these operations and the theory of linear transforma-tions to the solution of systems of linear equations. As a consequence of objective 1, we obtain a simple method for com-puting the rank of a linear transformation between finite-dimensional vector spaces by applying these rank-preserving matrix operations to a matrix that represents that transformation. Solving a system of linear equations is probably the most important ap-plication of linear algebra. The familiar method of elimination for solving systems of linear equations, which was discussed in Section 1.4, involves the elimination of variables so that a simpler system can be obtained. The tech-nique by which the variables are eliminated utilizes three types of operations: 1. interchanging any two equations in the system; 2. multiplying any equation in the system by a nonzero constant; 3. adding a multiple of one equation to another. In Section 3.3, we express a system of linear equations as a single matrix equation. In this representation of the system, the three operations above are the 'elementary row operations' for matrices. These operations provide a convenient computational method for determining all solutions to a system of linear equations. 147 148 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 3.1 ELEMENTARY MATRIX OPERATIONS AND ELEMENTARY MATRICES In this section, we define the elementary operations that are used throughout the chapter. In subsequent sections, we use these operations to obtain simple computational methods for determining the rank of a linear transformation and the solution of a system of linear equations. There are two types of el-ementary matrix operationsrow operations and column operations. As we will see, the row operations are more useful. They arise from the three opera-tions that can be used to eliminate variables in a system of linear equations. Definitions. Let A be an m x n matrix. Any one of the following three operations on the rows [columns] of A is called an elementary row [column] operation: (1) interchanging any two rows [columns] of A; (2) multiplying any row [column] of A by a nonzero scalar; (3) adding any scalar multiple of a row [column] of A to another row [col-umn]. Any of these three operations is called an elementary operation. Elemen-tary operations are of type 1, type 2, or type 3 depending on whether they are obtained by (1),12), or (3). Example 1 Let A = Interchanging the second row of A with the first row is an example of an elementary row operation of type 1. The resulting matrix is B = Multiplying the second column of A by 3 is an example of an elementary column operation of type 2. The resulting matrix is C = Sec. 3.1 Elementary Matrix Operations and Elementary Matrices 149 Adding 4 times the third row of A to the first row is an example of an elementary row operation of type 3. In this case, the resulting matrix is M = '17 2 7 12' 2 1 - 1 3 4 0 1 2 , Notice that if a matrix Q can be obtained from a matrix P by means of an elementary row operation, then P can be obtained from Q by an elementary row operation of the same type. (See Exercise 8.) So, in Example 1, A can be obtained from M by adding 4 times the third row of M to the first row ofM. Definition. An n x n elementary matrix is a matrix obtained by performing an elementary operation on In. The elementary matrix is said to be of type 1, 2, or 3 according to whether the elementary operation performed on In is a type 1, 2, or 3 operation, respectively. For example, interchanging the first two rows of I3 produces the elemen-tary matrix Note that E can also be obtained by interchanging the first two columns of J3. In fact, any elementary matrix can be obtained in at least two ways either by performing an elementary row operation on In or by performing an elementary column operation on In. (See Exercise 4.) Similarly, is an elementary matrix since it can be obtained from 1$ by an elementary column operation of type 3 (adding 2 times the first column of I3 to the third column) or by an elementary row operation of type 3 (adding 2 times the third row to the first row). Our first theorem shows that performing an elementary row operation on a matrix is equivalent to multiplying the matrix by an elementary matrix. Theorem 3.1. Let A MmXn(F), and suppose that B is obtained from A by performing an elementary row [column] operation. Then there exists an m x m [n x n] elementary matrix E such that B EA [B = AE]. fn fact, E is obtained from fm [fn] by performing the same elementary row [column] operation as that which was performed on A to obtain B. Conversely, if E is 150 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations an elementary m x m [n x n] matrix, then EA [AE] is the matrix obtained from A by performing the same elementary row [column] operation as that which produces E from Im [In]. The proof, which we omit, requires verifying Theorem 3.1 for each type of elementary row operation. The proof for column operations can then be obtained by using the matrix transpose to transform a column operation into a row operation. The details are left as an exercise. (See Exercise 7.) The next example illustrates the use of the theorem. Example 2 Consider the matrices A and B in Example 1. In this case, B is obtained from A by interchanging the first two rows of A. Performing this same operation on J3, we obtain the elementary matrix E = Note that EA = B. In the second part of Example 1, C is obtained from A by multiplying the second column of A by 3. Performing this same operation on I4, we obtain the elementary matrix E = ( 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1/ Observe that AE = C. It is a useful fact that the inverse of an elementary matrix is also an elementary matrix. Theorem 3.2. Elementary matrices are invertible, and the inverse of an elementary matrix is an elementary matrix of the same type. Proof. Let E be an elementary nxn matrix. Then E can be obtained by an elementary row operation on fn. By reversing the steps used to transform J n into E, we can transform E back into fn. The result is that In can be obtained from E by an elementary row operation of the same type. By Theorem 3.1, there is an elementary matrix E such that EE = fn. Therefore, by Exercise 10 of Section 2.4, E is invertible and E~1 = E. I Sec. 3.1 Elementary Matrix Operations and Elementary Matrices EXERCISES 151 1. Label the following statements as true or false. (a) An elementary matrix is always square. The only entries of an elementary matrix are zeros and ones. The n x n identity matrix is an elementary matrix. The product of two n x n elementary matrices is an elementary matrix. The inverse of an elementary matrix is an elementary matrix. The sum of two nxn elementary matrices is an elementary matrix. The transpose of an elementary matrix is an elementary matrix. If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then B can also be obtained by performing an elementary column operation on A. If B is a matrix that can be obtained by performing an elemen-tary row operation on a matrix A, then A can be obtained by performing an elementary row operation on B. (b) (c) (d) (e) (f) (g) (h) (i) 2. Let A--= / l 0 3 / l 0 31 , B = ( 1 - 2 1 , and C ^ 0 - 2 - 2 VI - 3 1/ l - 3 1 Find an elementary operation that transforms A into B and an elemen-tary operation that transforms B into C. By means of several additional operations, transform C into I3. 3. Use the proof of Theorem 3.2 to obtain the inverse of each of the fol-lowing elementary matrices. / o 0 1 / l 0 0 / 1 0 0 (a) 0 1 0 (b) ( 0 3 0 (c) ( 0 1 0 1 0 0 / 0 0 1/ - 2 0 1/ 4. Prove the assertion made on page 149: Any elementary nxn matrix can be obtained in at least two ways -either by performing an elementary row operation on In or by performing an elementary column operation on fn. 5. Prove that E is an elementary matrix if and only if Et is. 6. Let A be an m x n matrix. Prove that if B can be obtained from A by an elementary row [column] operation, then Bl can be obtained from At by the corresponding elementary column [row] operation. 7. Prove Theorem 3.1. 152 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 8. Prove that if a matrix Q can be obtained from a matrix P by an elemen-tary row operation, then P can be obtained from Q by an elementary row operation of the same type. Hint: Treat each type of elementary row operation separately. 9. Prove that any elementary row [column] operation of type 1 can be obtained by a succession of three elementary row [column] operations of type 3 followed by one elementary row [column] operation of type 2. 10. Prove that any elementary row [column] operation of type 2 can be obtained by dividing some row [column] by a nonzero scalar. 11. Prove that any elementary row [column] operation of type 3 can be obtained by subtracting a multiple of some row [column] from another row [column]. 12. Let A be an m x n matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms A into an upper triangular matrix. 3.2 THE RANK OF A MATRIX AND MATRIX INVERSES In this section, we define the rank of a matrix. We then use elementary operations to compute the rank of a matrix and a linear transformation. The section concludes with a procedure for computing the inverse of an invertible matrix. Definition, ffA G Mmxn(F), we define the rank of A, denoted rank(^4), to be the rank of the linear transformation LA Fn > Fm . Many results about the rank of a matrix follow immediately from the corresponding facts about a linear transformation. An important result of this type, which follows from Fact 3 (p. 100) and Corollary 2 to Theorem 2.18 (p. 102), is that an n x n matrix is invertible if and only if its rank is n. Every matrix A is the matrix representation of the linear transformation L,4 with respect to the appropriate standard ordered bases. Thus the rank of the linear transformation LA is the same as the rank of one of its matrix representations, namely, A. The next theorem extends this fact to any ma-trix representation of any linear transformation defined on finite-dimensional vector spaces. Theorem 3.3. Let T: V > W be a linear transformation between finite-dimensional vector spaces, and let 0 and 7 be ordered bases for V and W, respectively. Then rank(T) = rank([T]^). Proof. This is a restatement of Exercise 20 of Section 2.4. | Sec. 3.2 The Rank of a Matrix and Matrix Inverses 153 Now that the problem of finding the rank of a linear transformation has been reduced to the problem of finding the rank of a matrix, we need a result that allows us to perform rank-preserving operations on matrices. The next theorem and its corollary tell us how to do this. Theorem 3.4. Let A be an m x n matrix, ff P and Q are invertible m x m and nxn matrices, respectively, then (a) rank(j4154 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Let 0 be the standard ordered basis for Fn. Then 0 spans Fn and hence, by Theorem 2.2 (p. 68), K(LA) = span(LA(/?)) = span ({LA(ei), LA(e2),..., U (e n ) } ) . But, for any j, we have seen in Theorem 2.13(b) (p. 90) that LA(CJ) = Aej = a, , where aj is the j t h column of A. Hence Thus R(LA) =span({a i ,o 2 , . . . , a n } ) . rank(A) = dim(R(L/i)) = dim(span({ai ,a2 , . . . ,a})). Example 1 Let Observe that the first and second columns of A are linearly independent and that the third column is a linear combination of the first two. Thus rank(.A) = cjim I span = 2. To compute the rank of a matrix A, it is frequently useful to postpone the use of Theorem 3.5 until A has been suitably modified by means of appro-priate elementary row and column operations so that the number of linearly independent columns is obvious. The corollary to Theorem 3.4 guarantees that the rank of the modified matrix is the same as the rank of A. One such modification of A can be obtained by using elementary row and col-umn operations to introduce zero entries. The next example illustrates this procedure. Example 2 Let A = If we subtract the first row of A from rows 2 and 3 (type 3 elementary row operations), the result is Sec. 3.2 The Rank of a Matrix and Matrix Inverses 155 If we now subtract twice the first column from the second and subtract the first column from the third (type 3 elementary column operations), we obtain It is now obvious that the maximum number of linearly independent columns of this matrix is 2. Hence the rank of A is 2. The next theorem uses this process to transform a matrix into a particu-larly simple form. The power of this theorem can be seen in its corollaries. Theorem 3.6. Let A be an m x n matrix of rank r. Then r < m, r < n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix D = Ir OX Q2 Oa where 0, 02, and O3 are zero matrices. Thus Du = 1 for i < r and Dij = 0 otherwise. Theorem 3.6 and its corollaries are quite important. Its proof, though easy to understand, is tedious to read. As an aid in following the proof, we first consider an example. Example 3 Consider the matrix A = /0 2 4 2 2 4 4 4 8 0 8 2 0 10 2 6 3 2 9 1/ By means of a succession of elementary row and column operations, we can transform A into a matrix D as in Theorem 3.6. We list many of the inter-mediate matrices, but on several occasions a matrix is transformed from the preceding one by means of several elementary operations. The number above each arrow indicates how many elementary operations are involved. Try to identify the nature of each elementary operation (row or column and type) in the following matrix transformations. A) 2 4 2 2 4 4 4 8 0 8 2 0 10 2 6 3 2 9 1/ (4 4 4 8 0 0 2 4 2 2 8 2 0 10 2 6 3 2 9 1/ / 1 1 1 2 0 0 2 4 2 2 8 2 0 10 2 6 3 2 9 1/ 156 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (1 0 0 -[p -( 0 0 ^ 0 1 -6 -3 0 1 0 0 (I 0 0 Vo 0 2 - 8 - 4 0 0 1 2 1 1 2 > 2 4 2 2 - 6 - 8 - 6 2 - 3 - 4 - 3 1/ 0 0 A 1 1 - 6 2 2 ^ 0 0 - 3 1/ 0 0 0 / l c 0 0 0 2 0 4y l 0 1 0 0 v ( 3 0 1 0 0 0 0 1 0 / l 0 0 0 0 2 4 2 0 - 6 - 8 - 6 0 - 3 - 4 - 3 0 0 0 / l 2 1 1 4 0 8 3 * 0 0 2 0 4 / 0 0 0 / l 0 0 0 0 2 ( > I 0 1 0 0 V 0 2 2 i / 0 0 1 0 0 4 0 2 0 0 0 0 1 0 0 0 1 0 0 0 0 0 8 0 4 / Sec. 3.2 The Rank of a Matrix and Matrix Inverses 157 operation (each of type 1), we can move the nonzero entry to the 1,1 position (just as was done in Example 3). By means of at most one additional type 2 operation, we can assure a 1 in the 1,1 position. (Look at the second operation in Example 3.) By means of at most m 1 type 3 row operations and at most n 1 type 3 column operations, we can eliminate all nonzero entries in the first row and the first column with the exception of the 1 in the 1,1 position. (In Example 3, we used two row and three column operations to do this.) Thus, with a finite number of elementary operations, A can be transformed into a matrix B = where B' is an (m 1) x (n 1) matrix. In Example 3, for instance, / 2 4 2 2' B' =1-6 - 8 - 6 2 1 - 3 - 4 - 3 1 / 1 0 V o 0 0 B' ) By Exercise 11, B' has rank one less than B. Since rank(.A) = rank(jB) = r, rank(-B') r 1. Therefore r 1 < m 1 and r < n 1 by the induction hypothesis. Hence r < m and r < n. Also by the induction hypothesis, B' can be transformed by a finite num-ber of elementary row and column operations into the (m 1) x (n 1) matrix D' such that D' = / r -] On 04 Oe where O4, O5, and OQ are zero matrices. That is, D' consists of all zeros except for its first r 1 diagonal entries, which are ones. Let D = We see that the theorem now follows once we show that D can be obtained from B by means of a finite number of elementary row and column operations. However this follows by repeated applications of Exercise 12. Thus, since A can be transformed into B and B can be transformed into D, each by a finite number of elementary operations, A can be transformed into D by a finite number of elementary operations. / 1 0 o 0 D' 0 / 158 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Finally, since D' contains ones as its first r 1 diagonal entries, D contains ones as its first r diagonal entries and zeros elsewhere. This establishes the theorem. I Corollary 1. Let A be an m x n matrix of rank r. Then there exist invertible matrices B and C of sizes mxm and nxn, respectively, such that D = BAC, where D = fr o2 Ol o. is the mxn matrix in which Oi, 02, and O3 are zero matrices. Proof. By Theorem 3.6, A can be transformed by means of a finite number of elementary row and column operations into the matrix D. We can appeal to Theorem 3.1 (p. 149) each time we perform an elementary operation. Thus there exist elementary mxm matrices E,E2,... ,EP and elementary nxn matrices Gi,G2,...,Gq such that D EpEp-i E2EAGG2 Gq. By Theorem 3.2 (p. 150), each Ej and Gj is invertible. Let B EpEp- E and C = GG2- -Gq. Then B and C are invertible by Exercise 4 of Sec-tion 2.4, and D = BAC. I Corollary 2. Let A be an m x n matrix. Then (a) rank(4*) = rank(^l). (b) The rank of any matrix equals the maximum number of its linearly independent rows; that is, the rank of a matrix is the dimension of the subspace generated by its rows. (c) The rows and columns of any matrix generate subspaces of the same dimension, numerically equal to the rank of the matrix. Proof, (a) By Corollary 1, there exist invertible matrices B and C such that D = BAC, where D satisfies the stated conditions of the corollary. Taking transposes, we have Dl = (BAC)1 = CtAtBt. Since B and C are invertible, so are Bl and C* by Exercise 5 of Section 2.4. Hence by Theorem 3.4, rank(A') = rank(CtA*5*) = rank(Df). Suppose that r = rank(A). Then D* is an n x m matrix with the form of the matrix D in Corollary 1, and hence rank(Z)t) = r by Theorem 3.5. Thus rank(^4) = rank(Dt) =r = rank(^l). This establishes (a). The proofs of (b) and (c) are left as exercises. (See Exercise 13.) J^ Sec. 3.2 The Rank of a Matrix and Matrix Inverses 159 Corollary 3. Every invertible matrix is a product of elementary matrices. Proof. If A is an invertible nxn matrix, then rank(^4) = n. Hence the matrix D in Corollary 1 equals In, and there exist invertible matrices B and C such that In - BAC. As in the proof of Corollary 1, note that B = EpEp- E and C = GiG2-Gq, where the E^s and GVs are elementary matrices. Thus A = B-lInC-1 = B-lC~l, so that A = E^E^..-E-lG-1G-l1---G-[1. The inverses of elementary matrices are elementary matrices, however, and hence A is the product of elementary matrices. 1 We now use Corollary 2 to relate the rank of a matrix product to the rank of each factor. Notice how the proof exploits the relationship between the rank of a matrix and the rank of a linear transformation. Theorem 3.7. Let T: V > W and U: W Z be linear transformations on finite-dimensional vector spaces V, W, and Z, and let A and B be matrices such that the product AB is defined. Then (a) rank(UT) < rank(U). (b) rank(UT) < rank(T). / (c) rank(,4) < rank(A). (d) rank(AB) < rank(B). Proof. We prove these items in the order: (a), (c), (d), and (b). (a) Clearly, R(T) C W. Hence R(UT) = UT(V) = U(T(V)) = U(R(T)) C U(W) = R(U). Thus rank(UT) = dim(R(UT)) < dim(R(U)) = rank(U). (c) By (a), rank(AB) rank(LAs) = rank(L^LB) < rank(L^) = rank(^l). (d) By (c) and Corollary 2 to Theorem 3.6, mnk(AB) = rank((4)*) = rank(B tA t) < rank(Bf) = rank(B). (b) Let ct,0, and 7 be ordered bases for V, W, and Z, respectively, and let A' = [U]J and B' = [T]g. Then A'B' = [UT]2 by Theorem 2.11 (p. 88). Hence, by Theorem 3.3 and (d), rank(UT) = r a n k ^ ' S ' ) < rank(5') = rank(T). I J 160 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations It is important to be able to compute the rank of any matrix. We can use the corollary to Theorem 3.4, Theorems 3.5 and 3.6, and Corollary 2 to Theorem 3.6 to accomplish this goal. The object is to perform elementary row and column operations on a matrix to 'simplify' it (so that the transformed matrix has many zero entries) to the point where a simple observation enables us to determine how many linearly independent rows or columns the matrix has, and thus to determine its rank. Example 4 (a) Let A = 1 2 1 1 1 1 - 1 1 Note that the first and second rows of A are linearly independent since one is not a multiple of the other. Thus raxfk(A) 2. (b) Let / l 3 1 1 A- 1 0 1 1 0 3 0 0, In this case, there are several ways to proceed. Suppose that we begin with an elementary row operation to obtain a zero in the 2,1 position. Subtracting the first row from the second row, we obtain 0 0 3 - 3 3 1 1 0 0 0 0 Now note that the third row is a multiple of the second row, and the first and second rows are linearly independent. Thus rank(vl) = 2. As an alternative method, note that the first, third, and fourth columns of A are identical and that the first and second columns of A are linearly independent. Hence rank(v4) = 2. (c) Let A = Using elementary row operations, we can transform A as follows: 1 2 3 r 0 - 3 - 5 - 1 0 0 3 0, Sec. 3.2 The Rank of a Matrix and Matrix Inverses 161 It is clear that the last matrix has three linearly independent rows and hence has rank 3. In summary, perform row and column operations until the matrix is sim-plified enough so that the maximum number of linearly independent rows or columns is obvious. The Inverse of a Matrix We have remarked that an nxn matrix is invertible if and only if its rank is n. Since we know how to compute the rank of any matrix, we can always test a matrix to determine whether it is invertible. We now provide a simple technique for computing the inverse of a matrix that utilizes elementary row operations. Definition. Let A and B be m x n and m x p matrices, respectively. By the augmented matrix (AB), we mean the m x (n + p) matrix (A B), that is, the matrix whose first n columns arc the columns of A, and whose last p columns are the columns of B. Let A be an invertible nxn matrix, and consider the n x 2n augmented matrix C (AIn). By Exercise 15, we have A~lC = (A~lAA'lfn) = (InA~l). (1) By Corollary 3 to Theorem 3.6, A-1 is the product of elementary matrices, say A'1 = EpEp-i E. Thus (1) becomes EpEp_x Ei(AIn) = A~lC = (InA - i Because multiplying a matrix on the left by an elementary matrix transforms the matrix by an elementary row operation (Theorem 3.1 p. 149), we have the following result: If A is an invertible n x n matrix, then it is possible to transform the matrix (AIn) into the matrix (InA~x) by means of a finite number of elementary row operations. Conversely, suppose that A is invertible and that, for some nxn matrix B, the matrix (AIn) can be transformed into the matrix (InB) by a finite number of elementary row operations. Let E, E2,..., Ep be the elementary matrices associated with these elementary row operations as in Theorem 3.1; then EpEp-1---Ei(AIn) = {InB). Letting M = EpEp- E, we have from (2) that (MAM) = M(AIn) = (InB). (2) 162 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Hence MA = In and M = B. It follows that M = A'1. So B = A~ Thus we have the following result: If A is an invertible nxn matrix, and the matrix (AIn) is transformed into a matrix of the form (InB) by means of a finite number of elementary row operations, then B A~x. If, on the other hand, A is an n x n matrix that is not invertible, then rank(^4) < n. Hence any attempt to transform (^4|/n) into a matrix of the form (InB) by means of elementary row operations must fail because oth-erwise A can be transformed into In using the same row operations. This is impossible, however, because elementary row operations preserve rank. In fact, A can be transformed into a matrix with a row containing only zero entries, yielding the following result: If A is an n x n matrix that is not invertible, then any attempt to transform (AIn) into a matrix of the form (InB) produces a row whose first n entries are zeros. The next two examples demonstrate these comments. Example 5 We determine whether the matrix A = is invertible, and if i t is, we compute its inverse. We attempt to use elementary row operations to transform '0 2 4 (AI) = ( 2 4 2 3 3 1 into a matrix of the form (IB). One method for accomplishing this transfor-mation is to change each column of A successively, beginning with the first column, into the corresponding column of I. Since we need a nonzero entry in the 1,1 position, we begin by interchanging rows 1 and 2. The result is 2 0 3 4 2 3 2 4 1 0 1 0 1 0 0 0 0 1 In order to place a 1 in the 1,1 position, we must multiply the first row by ; this operation yields 0 3 2 2 3 1 4 1 0 1 0 1 0 0 0 0 1 Sec. 3.2 The Rank of a Matrix and Matrix Inverses 163 We now complete work in the first column by adding 3 times row 1 to row 3 to obtain ( 0 2 1 2 4 -3 - 2 0 0 - * 1 In order to change the second column of the preceding matrix into the second column of / , we multiply row 2 by | to obtain a 1 in the 2,2 position. This operation produces fl 0 Vo 2 1 - 3 1 2 - 2 o o o - i l We now complete our work on the second column by adding 2 times row 2 to row 1 and 3 times row 2 to row 3. The result is / l 0 0 1 0 0 - 1 0 0 0 6 _3 -i 2 2 ' 7 / Only the third column remains to be changed. In order to place a 1 in the 3,3 position, we multiply row 3 by this operation yields / l 0 0 1 0 0 - 3 2 1 5 ON 0 0 1 Adding appropriate multiples of row 3 to rows 1 and 2 completes the process and gives Thus A is invertible, and 0 0 IN i 2 1 V A' 3 = ( * _ i 4 3 ^ 8 2 x 4 i 164 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Example 6 We determine whether the matrix is invertible, and if it is, we compute its inverse. Using a strategy similar to the one used in Example 5, we attempt to use elementary row operations to transform (AI) = / I 2 2 1 1 5 1 - 1 4 1 0 0 0 1 0 0 0 1 into a matrix of the form (IB). We first add 2 times row 1 to row 2 and 1 times row 1 to row 3. We then add row 2 to row 3. The result, 1 2 2 1 1 5 1 - 1 4 1 0 0 1 0 0 0 0 1 1 0 0 2 - 3 3 1 - 3 3 / is a matrix with a row whose first 3 entries are zeros. Therefore A is not invertible. 1 0 0 2 - 3 0 1 - 3 0 1 - 2 - 1 1 - 2 - 3 0 1 0 0 1 1 0 0 1 0 0 1 Being able to test for invertibility and compute the inverse of a matrix allows us, with the help of Theorem 2.18 (p. 101) and its corollaries, to test for invertibility and compute the inverse of a linear transformation. The next example demonstrates this technique. Example 7 Let T: P2(R) -> P2(R) be defined by T(f(x)) = f(x) + f'(x) + f'(x), where f'(x) and f'(x) denote the first and second derivatives of f(x). We use Corollary 1 of Theorem 2.18 (p. 102) to test T for invertibility and compute the inverse if T is invertible. Taking 0 to be the standard ordered basis of P2(R), we have / l 1 2N [T]/3 = 0 1 2 0 0 1 Sec. 3.2 The Rank of a Matrix and Matrix Inverses 165 Using the method of Examples 5 and 6, we can show that T]p is invertible with inverse T - 1 0' ( [ T ^ r = 0 1 - 2 . ,0 0 1 Thus T is invertible, and ([T]^)'1 = [T -1]^. Hence by Theorem 2.14 (p. 91), we have [T l (a0 + aix + a2x2)]p = 0 Therefore T (ao + aix -I- a2x ) = (a0 ai) + (i - 2a2)x + a2x' / EXERCISES 1. Label the following statements as true or false. (a) The rank of a matrix is equal to the number of its nonzero columns. (b) The product of two matrices always has rank equal to the lesser of the ranks of the two matrices. (c) The mxn zero matrix is the only mxn matrix having rank 0. (d) Elementary row operations preserve rank. (e) Elementary column operations do not necessarily preserve rank. (f) The rank of a matrix is equal to the maximum number of linearly independent rows in the matrix. (g) The inverse of a matrix can be computed exclusively by means of elementary row operations. (h) The rank of an n x n matrix is at most n. (i) An n x n matrix having rank n is invertible. 2. Find the rank of the following matrices. (a) (c) 1 0 2 1 1 4 166 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (d) (f) 1 2 1 2 4 2 (e) / l 2 3 1 1 1 4 0 1 2 0 2 - 3 0 1 1 0 0 0 0/ / 1 2 3 1 - 4 2 0 4 1 6 2 - 8 1 1 l 3 0 5 1 - 3 l) (g) (I 1 0 1 2 2 0 2 1 1 0 1 V i o V 3. Prove that for any mxn matrix A, rank(A) 0 if and only if A is the zero matrix. 4. Use elementary row and column operations to transform each of the following matrices into a matrix D satisfying the conditions of Theo-rem 3.6, and then determine the rank of each matrix. (a) (b) 5. For each of the following matrices, compute the rank and the inverse if it exists. / (a) 1 2 1 1 (g) / 1 2 - 2 V 3 6. For each of the following linear transformations T, determine whether T is invertible, and compute T - 1 if it exists. (a) T: P2(R) -* P2(R) defined by T(/(x)) = f'(x) + 2f'(x) - f(x). (b) T: P2(R) -+ P2(R) defined by T(/(*)) = (x + l)f'(x). (c) T: R3 -> R3 defined by T(a,a2,a2,) = (ai + 2a2 + as, -a + a2 + 2a-s,ai + a3). Sec. 3.2 The Rank of a Matrix and Matrix Inverses (d) T: R3 - P2(R) defined by T(oi,02,03) = (a] + o2 + o3) + (ai - a2 + 03)3; + oix2 . (e) T: Pa(H) - R3 defined by T(/(x)) = ( / ( - l ) , / ( 0 ) , / ( l ) ) . (f) T: M2X2(-R) -> R4 defined by 167 T(A) = (tY(A),tr(At),tr(EA),tr(AE)), where E = 0 1 1 0 7. Express the invertible matrix ( ! ) as a product of elementary matrices. 8. Let A be an m x n matrix. Prove that if c is any nonzero scalar, then rank(c,4) = rank(^4). 9. Complete the proof of the corollary to Theorem 3.4 by showing that elementary column operations preserve rank. 10. Prove Theorem 3.6 for the case that A is an m x 1 matrix. 11. Let B = where B' is an m x n submatrix of B. Prove that if rank(B) = r, then rank(') = r - 1. 12. Let B' and D ' b e m x n matrices, and let B and D be (ra-f- 1) x (n + 1) matrices respectively defined by / ' 0 V 0 0 0 B' 7 B = ( 1 0 U 0 0 B' 7 and D / 1 0 U 0 0 D' 7 Prove that if B' can be transformed into D' by an elementary row [column] operation, then B can be transformed into D by an elementary row [column] operation. 168 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 13. Prove (b) and (c) of Corollary 2 to Theorem 3.6. 14. Let T, U: V > W be linear transformations. (a) Prove that R(T + U) C R(T) + R(U). (See the definition of the sum of subsets of a vector space on page 22.) (b) Prove that if W is finite-dimensional, then rank(T+U) < rank(T) + rank(U). (c) Deduce from (b) that rank(.A + B) < rank(A) + rank(5) for any mxn matrices A and B. 15. Suppose that A and B are matrices having n rows. M(AB) = (MAMB) for any mxn matrix M. Prove that 16. Supply the details to the proof of (b) of Theorem 3.4. 17. Prove that if B is a 3 x 1 matrix and C is a 1 x 3 matrix, then the 3 x 3 matrix BC has rank at most 1. Conversely, show that if A is any 3 x 3 matrix having rank 1, then there exist a 3 x 1 matrix B and a 1 x 3 matrix C such that A = BC. 18. Let A be an m x n matrix and B be an n x p matrix. Prove that AB can be written as a sum of n matrices of rank at most one. 19. Let A be an m x n matrix with rank m and B be an n x p matrix with rank n. Determine the rank of AB. Justify your answer. 20. Let / A = (a) Find a 5 x 5 matrix M with rank 2 such that AM = O, where O is the 4 x 5 zero matrix. (b) Suppose that B is a 5 x 5 matrix such that AB = O. Prove that rank(B) < 2. 21. Let A be an m x n matrix with rank m. Prove that there exists an nxm matrix B such that AB = Im. 22. Let B be an n x m matrix with rank m. Prove that there exists an mxn matrix A such that AB = Im. ( 1 - 1 - 2 3 0 1 1 - 1 - 1 3 4 - 5 2 - 1 - 1 1 l 0 3 - e ) 3.3 SYSTEMS OF LINEAR EQUATIONSTHEORETICAL ASPECTS This section and the next are devoted to the study of systems of linear equa-tions, which arise naturally in both the physical and social sciences. In this section, we apply results from Chapter 2 to describe the solution sets of Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 169 systems of linear equations as subsets of a vector space. In Section 3.4, el-ementary row operations are used to provide a computational method for finding all solutions to such systems. The system of equations (S) On2.-i + a2x2 021^1 + 022-X2 omiXi + arn2x2 + ainXn = b + a2nxn = 62 1 U-itviyi'rt 'tn where aij and h (1 < i < rn and 1 < j < n) are scalars in a field F and Xi,x2,... ,xn are n variables taking values in F, is called a system of m linear equations in n unknowns over the field F. The mxn matrix A = 021 i2 22 ai a2l flml O-m'2 J is called the coefficient matrix of the system (S). If we let x x2 xn) and b = b2 then the system (S) may be rewritten as a single matrix equation Ax = b. To exploit the results that we have developed, we often consider a system of linear equations as a single matrix equation. A solution to the system (S) is an n-tuple s2 SnJ F' such that As = b. The set of all solutions to the system (S) is called the solution set of the system. System (S) is called consistent if its solution set is nonempty; otherwise it is called inconsistent. 170 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Example 1 (a) Consider the system xi + x2 = 3 X x2 = 1. By use of familiar techniques, we can solve the preceding system and conclude that there is only one solution: xi 2, x2 = 1; that is, 1 / ' In matrix form, the system can be written 1 1 (xi 1 - 1 / X2 1) so A = l _ J ) and b=(3.. (b) Consider / 2rci + 3x2 + x3 = 1 x x2 + 2z3 = 6; that is, 2 3 1 1 - 1 2 'xi X2 ,X3 This system has many solutions, such as ' -6N s 2 | and s 6/ -4 | (c) Consider that is, xi + x2 = 0 xi +x2 = 1; 1 1 /a* 1 1/ U 2 0^ It is evident that this system has no solutions. Thus we see that a system of linear equations can have one, many, or no solutions. Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 171 We must be able to recognize when a system has a solution and then be able to describe all its solutions. This section and the next are devoted to this end. We begin our study of systems of linear equations by examining the class of homogeneous systems of linear equations. Our first result (Theorem 3.8) shows that the set of solutions to a homogeneous system of m linear equations in n unknowns forms a subspace of Fn. We can then apply the theory of vector spaces to this set of solutions. For example, a basis for the solution space can be found, and any solution can be expressed as a linear combination of the vectors in the basis. Definitions. A system Ax = b of m linear equations in n unknowns is said to be homogeneous if b = 0. Otherwise the system is said to be nonhomogeneous. Any homogeneous system has at least one solution, namely, the zero vec-tor. The next result gives further information about the set of solutions to a homogeneous system. Theorem 3.8. Let Ax = 0 be a homogeneous system of m linear equa-tions in n unknowns over a field F. Let K denote the set of all solutions to Ax = 0. Then K = N(L^); hence K is a subspace of Fn of dimension n rank(L/i) = n rank(A). Proof. Clearly, K = {s G Fn : As = 0} = N(LA). The second part now follows from the dimension theorem (p. 70). 1 Corollary. Ifrn < n, the system Ax = 0 has a nonzero solution. Proof. Suppose that m < n. Then rank(A) = rank(L/i) < m. Hence dim(K) = n rank(Lyi) > n m > 0, where K = N(L/). Since dim(K) > 0, K ^ {()}. Thus there exists a nonzero vector s K; so s is a nonzero solution to Ax 0. 1 Example 2 (a) Consider the system Xi + 2.T2 + x3 = 0 x, - x2 - x:i = 0. Let A = 1 2 1 1 - 1 - 1 172 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations be the coefficient matrix of this system. It is clear that rank(^l) = 2. If K is the solution set of this system, then dim(K) = 3 2 1. Thus any nonzero solution constitutes a basis for K. For example, since is a solution to the given system, is a basis for K. Thus any vector in K is of the form where t G R. (b) Consider the system X 2x2 + x$ 0 of one equation in three unknowns. If A = (l 2 l) is the coefficient matrix, then rank (A) = 1. Hence if K is the solution set, then dim(K) = 3 1 = 2. Note that / 2 / -1N l j ana ( 0 are linearly independent vectors in K. Thus they constitute a basis for K, so that K = < i In Section 3.4, explicit computational methods for finding a basis for the solution set of a homogeneous system are discussed. We now turn to the study of nonhomogeneous systems. Our next result shows that the solution set of a nonhomogeneous system Ax = b can be described in terms of the solution set of the homogeneous system Ax = 0. We refer to the equation Ax = 0 as the homogeneous system corresponding to Ax = b. Theorem 3.9. Let K be the solution set of a system of linear equations Ax = b, and let KH be the solution set of the corresponding homogeneous system Ax 0. Then for any solution s to Ax b K = {,s} + KH = {,s + fc:fcGKH}. Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 173 Proof. Let s be any solution to Ax = b. We must show that K = {.'?} + KH. If w G K, then Aw b. Hence A(w - s) = Aw - As = b- b= 0. So w s G KH. Thus there exists k G KH such that w s k. It follows that w = s + fc G {S} + KH, and therefore KC{S} + KH-Conversely, suppose that w G {s} + KH; then w = s + k for some fce KH. But then Aw = A(s + k) = As + Ak = b + 0 = b; so w G K. Therefore {} + KH QK, and thus K-{s} + K. I Example 3 (a) Consider the system Xi + 2x2 + 3 = 7 #1 #2 X3 = 4. The corresponding homogeneous system is the system in Example 2(a). It is easily verified that ' ( I ) ' is a solution to the preceding nonhomogeneous system. So the solution set of the system is K = { ( i ) + t ( t l : t e * by Theorem 3.9. (b) Consider the system x 2x2 + x% 4. The corresponding homoge-neous system is the system in Example 2(b). Since is a solution to the given system, the solution set K can be written as K; , | 1 j +* 2 ( o ) /,./,, R. 0 174 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations The following theorem provides us with a means of computing solutions to certain systems of linear equations. Theorem 3.10. Let Ax = b be a system of n linear equations in n unknowns. If A is invertible, then the system has exactly one solution, namely, A~lb. Conversely, if the system has exactly one solution, then A is invertible. Proof. Suppose that A is invertible. Substituting A~lb into the system, we have A(A~lb) = (AA~x)b = b. Thus A~xb is a solution. If s is an arbitrary solution, then As = b. Multiplying both sides by A~l gives s A~1b. Thus the system has one and only one solution, namely, A~lb. Conversely, suppose that the system has exactly one solution s. Let KH denote the solution set for the corresponding homogeneous system Ax = 0. By Theorem 3.9, {s} = {s} + KH. But this is so only if KH = {0}. Thus N(L^) = {()}, and hence A is invertible. jif Example 4 Consider the following system of three linear equations in three unknowns: 2x2 + 4x3 = 2 2xi + 4.T2 + 2x3 = 3 3xi + 3x2 + X3 = 1. In Example 5 of Section 3.2, we computed the inverse of the coefficient matrix A of this system. Thus the system has exactly one solution, namely, * A 1 X2 w , = A lb = ( * 4 3 V 8 5 8 4 3 8 4 ' 2 1 f2 I 3 U w (-l 4 l - M We use this technique for solving systems of linear equations having in-vertible coefficient matrices in the application that concludes this section. In Example 1(c), we saw a system of linear equations that has no solutions. We now establish a criterion for determining when a system has solutions. This criterion involves the rank of the coefficient matrix of the system Ax b and the rank of the matrix (>1|&). The matrix (^4|6) is called the augmented matrix of the system Ax = b. Theorem 3.11. Let Ax = b be a system of linear equations. Then the system is consistent if and only ifrank(A) = rank(-A|fr). Proof. To say that Ax = b has a solution is equivalent to saying that b G R(L/i). (See Exercise 9.) In the proof of Theorem 3.5 (p. 153), we saw that R(Lyi) =span({a i ,o 2 , . . . ,o}) , Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 175 the span of the columns of A. Thus Ax = b has a solution if and only if b G span({ai,02,.. . ,an}). But b G span({oi,02,.. . ,on}) if and only if span({ai,02,.. . ,on}) = span({ai,a2,... ,an,b}). This last statement is equivalent to dim(span({oi, o 2 , . . . , on})) = dim(span({oi, a2,..., an, b})). So by Theorem 3.5, the preceding equation reduces to rank(v4) = rank(A|6). I Example 5 Recall the system of equations xi + x2 0 xi + x2 = 1 in Example 1(c). Since A = 1 1 1 1 and (Ab) = 1 1 0 1 1 1 rank(A) = 1 and rank(A|o) = 2. Because the two ranks are unequal, the system has no solutions. Example 6 We can use Theorem 3.11 to determine whether (3,3,2) is in the range of the linear transformation T: R3 * R3 defined by T(oi ,a2 ,o3) = (ai +0.2 +03,01 - o2 + a 3 , a i +o 3 ) . Now (3,3,2) G R(T) if and only if there exists a vector s = (xi,X2,xa) in R3 such that T($) (3,3,2). Such a vector s must be a solution to the system x + x2 + x3 = 3 X] X2 + X3 = 3 X + X3 = 2. Since the ranks of the coefficient matrix and the augmented matrix of this system are 2 and 3, respectively, it follows that this system has no solutions. Hence (3,3,2) R(T). 176 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations An Application In 1973, Wassily Leontief won the Nobel prize in economics for his work in developing a mathematical model that can be used to describe various economic phenomena. We close this section by applying some of the ideas we have studied to illustrate two special cases of his work. We begin by considering a simple society composed of three people (industries)-a farmer who grows all the food, a tailor who makes all the clothing, and a carpenter who builds all the housing. We assume that each person sells to and buys from a central pool and that everything produced is consumed. Since no commodities either enter or leave the system, this case is referred to as the closed model. Each of these three individuals consumes all three of the commodities pro-duced in the society. Suppose that the proportion of each of the commodities consumed by each person is given in the following table. Notice that each of the columns of the table must sum to 1. Food Clothing Housing Farmer Tailor Carpenter 0.40 0.10 0.50 0.20 0.70 0.10 0.20 0.20 0.60 Let Pi,p2, and p% denote the incomes of the farmer, tailor, and carpenter, respectively. To ensure that this society survives, we require that the con-sumption of each individual equals his or her income. Note that the farmer consumes 20% of the clothing. Because the total cost of all clothing is p2, the tailor's income, the amount spent by the farmer on clothing is 0.20p2-Moreover, the amount spent by the farmer on food, clothing, and housing must equal the farmer's income, and so we obtain the equation 0.40pi + 0.20p2 + 0.20p3 = Pi-Similar equations describing the expenditures of the tailor and carpenter pro-duce the following system of linear equations: 0.40pi + 0.20p2 + 0.20p3 =7 Pi O.lOpi + 0.70p2 + 0.20p3 = Pi 0.50pi + 0.10p2 + 0.60p3 = Pa-This system can be written as Ap = p, where P = Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 177 and A is the coefficient matrix of the system. In this context, A is called the input-output (or consumption) matrix, and Ap = p is called the equilibrium condition. For vectors 6 = (b,b2,... , bn) and c (c,c2,... , cn) in Rn, we use the notation b > c [b > c] to mean bi > c-i [bi > Ci] for all i. The vector b is called nonnegative [positive] if b > 0 [b > 0]. At first, it may seem reasonable to replace the equilibrium condition by the inequality Ap < p, that is, the requirement that consumption not exceed production. But, in fact, Ap < p implies that Ap p in the closed model. For otherwise, there exists a k for which pk >^AkjPj-j Hence, since the columns of A sum to 1, 5 3 > 5Z ^PS = ( J 2 AiA Pi = * * ' * 3 which is a contradiction. One solution to the homogeneous system (I A)x = 0, which is equivalent to the equilibrium condition, is / We may interpret this to mean that the society survives if the farmer, tailor, and carpenter have incomes in the proportions 25 : 35 : 40 (or 5 : 7 : 8). Notice that we are not simply interested in any nonzero solution to the system, but in one that is nonnegative. Thus we must consider the question of whether the system (/ A)x = 0 has a nonnegative solution, where A is a matrix with nonnegative entries whose columns sum to 1. A useful theorem in this direction (whose proof may be found in 'Applications of Matrices to Economic Models and Social Science Relationships,' by Ben Noble, Proceed-ings of the Summer Conference for College Teachers on Applied Mathematics, 1971, CUPM, Berkeley, California) is stated below. Theorem 3.12. Let A be an n x n input output matrix having the form A = B C D E where D is a l x ( n - l ) positive vector and C is an ( n - l ) x l positive vector. Then (I A)x = 0 has a one-dimensional solution set that is generated by a nonnegative vector. 178 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Observe that any input-output matrix with all positive entries satisfies the hypothesis of this theorem. The following matrix does also: '0.75 0.50 0.65' 0 0.25 0.35 v0.25 0.25 0 In the open model, we assume that there is an outside demand for each of the commodities produced. Returning to our simple society, let Xi,X2, and X3 be the monetary values of food, clothing, and housing produced with respective outside demands d,d2, and 0*3. Let A be the 3 x 3 matrix such that Aij represents the amount (in a fixed monetary unit such as the dollar) of commodity i required to produce one monetary unit of commodity j. Then the value of the surplus of food in the society is xi - (Auxx + A12X2 + ^13^3), that is, the value of food produced minus the value of food consumed while producing the three commodities. The assumption that everything produced is consumed gives us a similar equilibrium condition for the open model, namely, that the surplus of each of the three commodities must equal the corresponding outside demands. Hence .{' 1 / , AijXj 3=1 = di fori = 1,2,3. In general, we must find a nonnegative solution to ( / A)x = d, where A is a matrix with nonnegative entries such that the sum of the entries of each column of A does not exceed one, and d > 0. It is easy to see that if ( / A)-1 exists and is nonnegative, then the desired solution is ( / A)~ld. Recall that for a real number a, the series 1 + a + a2 + converges to (1 o) _ 1 if a < 1. Similarly, it can be shown (using the concept of conver-gence of matrices developed in Section 5.3) that the series I + A + A2 + converges to (/ A)'1 if {A71} converges to the zero matrix. In this case, (I .A)-1 is nonnegative since the matrices J, A, A2,... are nonnegative. To illustrate the open model, suppose that 30 cents worth of food, 10 cents worth of clothing, and 30 cents worth of housing are required for the production of $1 worth of food. Similarly, suppose that 20 cents worth of food, 40 cents worth of clothing, and 20 cents worth of housing are required for the production of $1 of clothing. Finally, suppose that 30 cents worth of food, 10 cents worth of clothing, and 30 cents worth of housing are required for the production of $1 worth of housing. Then the input output matrix is /0.30 0.20 0.30 A = I 0.10 0.40 0.10 ; 0.30 0.20 0.30/ Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 179 so I-A = 0.70 -0.20 -0.30' -0.10 0.60 -0.10 -0.30 -0.20 0.70, and (I^A)~1 = 2.0 0.5 1.0 1.0 2.0 1.0 1.0 0.5 2.0 Since (I A)'1 is nonnegative, we can find a (unique) nonnegative solution to (/ A)x d for any demand d. For example, suppose that there are outside demands for $30 billion in food, $20 billion in clothing, and $10 billion in housing. If we set d = then x = (I-A)~ld= 60 So a gross production of $90 billion of food, $60 billion of clothing, and $70 billion of housing is necessary to meet the required demands. EXERCISES 1. Label the following statements as true or false. (a) Any system of linear equations has at least one solution. (b) Any system of linear equations has at most one solution. (c) Any homogeneous system of linear equations has at least one so-lution. (d) Any system of n linear equations in n unknowns has at most one solution. (e) Any system of n linear equations in n unknowns has at least one solution. (f) If the homogeneous system corresponding to a given system of lin-ear equations has a solution, then the given system has a solution. (g) If the coefficient matrix of a homogeneous system of n linear equa-tions in n unknowns is invertible, then the system has no nonzero solutions. (h) The solution set of any system of m linear equations in n unknowns is a subspace of Fn. 2. For each of the following homogeneous systems of linear equations, find the dimension of and a basis for the solution set. 180 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (a) (c) xi + 3x2 = 0 2xi + 6x2 = 0 xi + 2x2 - x3 = 0 2xi + x2 + x3 = 0 (b) Xi -f X2 X3 = 0 4xi + X2 - 2x3 = 0 2xi + x2 - x3 = 0 (d) xi - x2 + x 3 = 0 xi + 2x2 - 2x3 = 0 (e) xi + 2x2 - 3x3 + x4 = 0 Xi + 2X2 + X3 + X4 = 0 (f) xi + 2x2 = 0 Xi x2 = 0 (g) X2 X3 + X4 = 0 3. Using the results of Exercise 2, find all solutions to the following sys-tems. (a) (c) (e) xi + 3x2 = 5 2xi + 6x2 = 10 Xi + 2X2 X3 = 3 2xi + x2 + x3 = 6 xi + 2x2 -^3x3 + x4 = 1 (b) Xl + X2 - X3 = 1 4xi + x2 - 2x3 = 3 2xi + x2 - x3 = 5 (d) Xi - X2 + X3 as 1 xi + 2x2 - 2x3 = 4 (f) xi + 2x2 = 5 Xi - x2 = - 1 (g) Xi + 2X2 + X3 + X4 = 1 X2 ~ X3 + X4 = 1 4. For each system of linear equations with the invertible coefficient matrix A, (1) Compute A'1. (2) Use A-1 to solve the system. W 2 x 1 + 5 x 2 = 3 M *l+*2ZX3 = 2xi - 2x2 + x 3 = 4 5. Give an example of a system of n linear equations in n unknowns with infinitely many solutions. 6. Let T: R3 R2 be defined by T(a, b, c) = (a + 6,2a c). Determine T - K u i ) -7. Determine which of the following systems of linear equations has a so-lution. . Sec. 3.3 Systems of Linear EquationsTheoretical Aspects 181 X -r x2 Xg = 1 2xi + x2 + 3x3 = 2 X + X2 + 3X3 X4 = 0 Xl + X2+ X3 + X4 = 1 Xi - 2x2 + X3 - X4 = 1 4xi + X2 + 8x3 X4 = 0 8. 9. Xi + X2 X3 + 2x4 = 2 (a) xi + x2 + 2x3 = 1 (b) 2xi + 2x2 + x3 + 2x4 = 4 xi + 2x2 + 3x3 = 1 (c) xi + x2 - x3 = 0 (d) xi -I- 2x2 + X3 = 3 Xi + 2x2 X3 = 1 (e) 2xi + x2 + 2x3 = 3 Xi 4x2 + 7x3 = 4 Let T: R3 -+ R3 be defined by T(o,6,c) = (o + 6,6 - 2c, o + 2c). For each vector v in R3, determine whether v R(T). (a) v = (1 ,3 , -2) (b) ^ = (2,1,1) Prove that the system of linear equations Ax = b has a solution if and only if 6 e R(L^). 10. Prove or give a counterexample to the following statement: If the co-efficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution. / 11. In the closed model of Leontief with food, clothing, and housing as the basic industries, suppose that the input-output matrix is A = At what ratio must the farmer, tailor, and carpenter produce in order for equilibrium to be attained? 12. A certain economy consists of two sectors: goods and services. Suppose that 60% of all goods and 30% of all services are used in the production of goods. What proportion of the total economic output is used in the production of goods? 13. In the notation of the open model of Leontief, suppose that (& 5 16 1 1 4 1 2 1 6 1 3 ft 5 16 1 2J A = and d = are the input-output matrix and the demand vector, respectively. How much of each commodity must be produced to satisfy this demand? 182 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 14. A certain economy consisting of the two sectors of goods and services supports a defense system that consumes $90 billion worth of goods and $20 billion worth of services from the economy but does not contribute to economic production. Suppose that 50 cents worth of goods and 20 cents worth of services are required to produce $1 worth of goods and that 30 cents worth of goods and 60 cents worth of services are required to produce $1 worth of services. What must the total output of the economic system be to support this defense system? 3.4 SYSTEMS OF LINEAR EQUATIONS-COMPUTATIONAL ASPECTS In Section 3.3, we obtained a necessary and sufficient condition for a system of linear equations to have solutions (Theorem 3.11 p. 174) and learned how to express the solutions to a nonhomogeneous system in terms of solutions to the corresponding homogeneous system (Theorem 3.9 p. 172). The latter result enables us to determine all the solutions to a given system if we can find one solution to the given system and a basis for the solution set of the corresponding homogeneous system. In this section, we use elementary row operations to accomplish these two objectives simultaneously. The essence of this technique is to transform a given system of linear equations into a system having the same solutions, but which is easier to solve (as in Section 1.4). Definition. Two systems of linear equations are called equivalent if they have the same solution set. The following theorem and corollary give a useful method for obtaining equivalent systems. Theorem 3.13. Let Ax = 6 be a system of m linear equations in n unknowns, and let C be an invertible mxm matrix. Then the system (CA)x = Cb is equivalent to Ax b. Proof. Let K be the solution set for Ax = 6 and K' the solution set for (CA)x = Cb. If w Sec. 3.4 Systems of Linear EquationsComputational Aspects 183 Proof. Suppose that (A'b') is obtained from (^4|6) by elementary row operations. These may be executed by multiplying (Ab) by elementary mx m matrices E, E2,..., Ep. Let C = Ep E2E; then (A'b') = C(Ab) = (CACb). Since each Ei is invertible, so is C. Now A' = CA and b' = Cb. Thus by Theorem 3.13, the system A'x = b' is equivalent to the system Ax = b. I We now describe a method for solving any system of linear equations. Consider, for example, the system of linear equations 3xi + 2x2 + 3x3 - 2x4 = 1 xi + x2 + x3 = 3 xi + 2x2 + x3 - x4 = 2. First, we form the augmented matrix 3 1 1 2 3 1 1 2 1 - 2 0 - 1 By using elementary row operations, we transform the/augmented matrix into an upper triangular matrix in which the first nonzero entry of each row is 1, and it occurs in a column to the right of the first nonzero entry of each preceding row. (Recall that matrix A is upper triangular if Aij 0 whenever i>h) 1. In the leftmost nonzero column, create a 1 in the first row. In our example, we can accomplish this step by interchanging the first and third rows. The resulting matrix is 1 2 1 1 1 1 3 2 3 - 1 0 - 2 2. By means of type 3 row operations, use the first row to obtain zeros in the remaining positions of the leftmost nonzero column. In our example, we must add 1 times the first row to the second row and then add 3 times the first row to the third row to obtain 0 0 2 1 - 1 0 - 4 0 - 1 1 1 2 1 - 5 3. Create a 1 in the next row in the leftmost possible column, without using previous row(s). In our example, the second column is the leftmost 184 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations possible column, and we can obtain a 1 in the second row, second column by multiplying the second row by 1. This operation produces 0 0 2 1 1 0 - 4 0 - 1 - 1 1 2 - 1 - 5 4. Now use type 3 elementary row operations to obtain zeros below the 1 created in the preceding step. In our example, we must add four times the second row to the third row. The resulting matrix is 0 0 2 1 1 0 0 0 - 1 - 1 - 3 5. Repeat steps 3 and 4 on each succeeding row until no nonzero rows remain. (This creates zeros above the first nonzero entry in each row.) In our example, this can be accomplished by multiplying the third row by | . This operation produces 1 0 0 2 1 1 0 0 0 - 1 - 1 1 2 - 1 3 We have now obtained the desired matrix. To complete the simplification of the augmented matrix, we must make the first nonzero entry in each row the only nonzero entry in its column. (This corresponds to eliminating certain unknowns from all but one of the equations.) 6. Work upward, beginning with the last nonzero row, and add multiples of each row to the rows above. (This creates zeros above the first nonzero entry in each row.) In our example, the third row is the last nonzero row, and the first nonzero entry of this row lies in column 4. Hence we add the third row to the first and second rows to obtain zeros in row 1, column 4 and row 2, column 4. The resulting matrix is 0 0 2 1 0 1 0 0 0 0 1 7. Repeat the process described in step 6 for each preceding row until it is performed with the second row, at which time the reduction process is complete. In our example, we must add 2 times the second row to the first row in order to make the first row, second column entry become zero. This operation produces 0 0 0 1 1 0 0 0 0 0 1 Sec. 3.4 Systems of Linear EquationsComputational Aspects 185 We have now obtained the desired reduction of the augmented matrix. This matrix corresponds to the system of linear equations x + x3 = 1 x2 = 2 X4 = 3. Recall that, by the corollary to Theorem 3.13, this system is equivalent to the original system. But this system is easily solved. Obviously X2 = 2 and X4 = 3. Moreover, xi and X3 can have any values provided their sum is 1. Letting X3 = t, we then have xi = 1 t. Thus an arbitrary solution to the original system has the form fl-t 2 t K ^ j = f 2 0 A + * f-l 0 1 V / Observe that 0 1 V 186 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations a new row, we must move to the right one or more columns to find the first nonzero entry of the new row. (b) The matrix is not in reduced row echelon form, because the first column, which contains the first nonzero entry in row 1, contains another nonzero entry. Similarly, the matrix '0 1 0 2y 1 0 0 1 ,0 0 1 1 , is not in reduced row echelon form, because the first nonzero entry of the second row is not to the right of the first nonzero entry of the first row. Finally, the matrix / is not in reduced row echelon form, because the first nonzero entry of the first row is not 1. It can be shown (see the corollary to Theorem 3.16) that the reduced row echelon form of a matrix is unique; that is, if different sequences of elementary row operations are used to transform a matrix into matrices Q and Q' in reduced row echelon form, then Q = Q'. Thus, although there are many different sequences of elementary row operations that can be used to transform a given matrix into reduced row echelon form, they all produce the same result. The procedure described on pages 183-185 for reducing an augmented matrix to reduced row echelon form is called Gaussian elimination. It consists of two separate parts. 1. In the forward pass (steps 1-5), the augmented matrix is transformed into an upper triangular matrix in which the first nonzero entry of each row is 1, and it occurs in a column to the right of the first nonzero entry of each preceding row. 2. In the backward pass or back-substitution (steps 6-7), the upper trian-gular matrix is transformed into reduced row echelon form by making the first nonzero entry of each row the only nonzero entry of its column. P Sec. 3.4 Systems of Linear EquationsComputational Aspects 187 Of all the methods for transforming a matrix into its reduced row ech-elon form, Gaussian elimination requires the fewest arithmetic operations. (For large matrices, it requires approximately 50% fewer operations than the Gauss-Jordan method, in which the matrix is transformed into reduced row echelon form by using the first nonzero entry in each row to make zero all other entries in its column.) Because of this efficiency, Gaussian elimination is the preferred method when solving systems of linear equations on a com-puter. In this context, the Gaussian elimination procedure is usually modified in order to minimize roundoff errors. Since discussion of these techniques is inappropriate here, readers who are interested in such matters are referred to books on numerical analysis. When a matrix is in reduced row echelon form, the corresponding sys-tem of linear equations is easy to solve. We present below a procedure for solving any system of linear equations for which the augmented matrix is in reduced row echelon form. First, however, we note that every matrix can be transformed into reduced row echelon form by Gaussian elimination. In the forward pass, we satisfy conditions (a) and (c) in the definition of reduced row echelon form and thereby make zero all entries below the first nonzero entry in each row. Then in the backward pass, we make zero all entries above the first nonzero entry in each row, thereby satisfying condition (b) in the definition of reduced row echelon form. Theorem 3.14. Gaussian elimination transforms any matrix into its re-duced row echelon form. We now describe a method for solving a system in which the augmented matrix is in reduced row echelon form. To illustrate this procedure, we con-sider the system 2xi + 3x2 + X3 + 4x4 9x5 = 17 xi + x2 + X3 + X4 3x5 = 6 xi 4- x2 + X3 + 2x4 - 5x5 = 8 2x, + 2x2 + 2x3 + 3x4 - 8x5 = 14, for which the augmented matrix is (2 3 1 4 1 1 1 1 1 V - 9 - 3 - 5 - 8 17 6 8 1 4 / Applying duces the Gaussian elimination to the augmented matrix of the system pro-following sequence of matrices. 17 6 8 UJ ( 2 1 V2 1 3 1 2 1 1 1 4 1 2 2 3 - 3 - 9 - 5 - 8 6 17 8 14, 188 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations / l 1 1 1 0 1 - 1 2 0 0 0 0 a i 0 1 0 0 0 0 - 3 - 3 0 1 - 2 0 1 - 2 6 5 2 V 5 2 y 1 0 -1 0 0 1 0 0 - 1 1 - 2 0 4^ 1 2 t2 = / / 3 1 0 2 w -rh f-2 1 1 0 I 0) + t2 ( 2 - 1 0 2 I 1/ where ti,t2 6 R. Notice that ( ! { f-2 1 1 0 I o^ 5 < 2 - 1 0 2 I V J Sec. 3.4 Systems of Linear EquationsComputational Aspects 189 is a basis for the solution set of the corresponding homogeneous system of equations and / 3 1 0 2 w is a particular solution to the original system. Therefore, in simplifying the augmented matrix of the system to reduced row echelon form, we are in effect simultaneously finding a particular solu-tion to the original system and a basis for the solution set of the associated homogeneous system. Moreover, this procedure detects when a system is in-consistent, for by Exercise 3, solutions exist if and only if, in the reduction of the augmented matrix to reduced row echelon form, we do not obtain a row in which the only nonzero entry lies in the last column. Thus to use this procedure for solving a system Ax = 6 of m linear equa-tions in n unknowns, we need only begin to transform the augmented matrix (.4|6) into its reduced row echelon form (^4'|6') by means of Gaussian elimi-nation. If a row is obtained in which the only nonzero entry lies in the last column, then the original system is inconsistent. Otherwise, discard any zero rows from (A'b'), and write the corresponding system of equations. Solve this system as described above to obtain an arbitrary solution of the form s = so + hui + t2u2 H + t n _ r w n _ r , where r is the number of nonzero rows in A' (r < m). The preceding equation is called a general solution of the system Ax = 6. It expresses an arbitrary solution s of Ax = b in terms of n r parameters. The following theorem states that s cannot be expressed in fewer than n r parameters. Theorem 3.15. Let Ax = 6 be a system of r nonzero equations in n unknowns. Suppose that rank(A) rank(^|6) and that (Ab) is in reduced row echelon form. Then (a) rank(A) = r. (b) If the general solution obtained by the procedure above is of the form s = s0 + hui + t2u2 H - tn-run-r, then {ui,u2,..., un-r} is a basis for the solution set of the correspond-ing homogeneous system, and So is a solution to the original system. Proof. Since (Ab) is in reduced row echelon form, (Ab) must have r nonzero rows. Clearly these rows are linearly independent by the definition of the reduced row echelon form, and so rank(yl|6) = r. Thus rank(yl) = r. 190 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations Let K be the solution set for Ax = b. and let KH be the solution set for Ax = 0. Setting t = t2 tn-r = 0, we see that s = So G K. But by Theorem 3.9 (p. 172), K = {.s,,} + KH. Hence KH = {so} + K = span({u 1 .u 2 , . . . . -,}) Because rank(A) = r, we have dim(Kn) = n r. Thus since dim(Kn) = n r and KH is generated by a set {u,u2,... ,un-r} containing at most n r vectors, we conclude that this set is a basis for KH- 1 An Interpretation of the Reduced Row Echelon Form Let A be an m x n matrix with columns Oi,a2, ,an< a i Jd let B be the reduced row echelon form of A. Denote the columns of B by b,b2,...,b. If the rank of A is r, then the rank of B is also r by the corollary to Theorem 3.4 (p. 153). Because B is in reduced row echelon form, no nonzero row of B can be a linear combination of the other rows of B. Hence B must have exactly r nonzero rows, and if r > 1, the vectors e., e2,..., er must occur among the columns of B. For i = 1 ,2 , . . . , r, let j.t denote a column number of B such that bjt = e. We claim that a7 l , aj2..... a J r , the columns of A corresponding to these columns of B, are linearly independent. For suppose that there are scalars c,c2,...,cr such that cUj1 + c2aj2 - 1- crajr = 0. Because B can be obtained from A by a sequence of elementary row oper-ations, there exists (as in the proof of the corollary to Theorem 3.13) an invertible mxm matrix M such that MA = B. Multiplying the preceding equation by M yields CIMOJ, +c2Ma 3i + crMa.; = 0. Since Ma*. = 6, = a, it follows that C(' + C2C2 H r Cr('r = 0. Hence c = c2 = = cr = 0. proving that the vectors ctj11Q>ja, ,o-jr are linearly independent. Because B has only r nonzero rows, every column of B has the form dr 0 W Sec. 3.4 Systems of Linear EquationsComputational Aspects for scalars d,d2,... ,dr. The corresponding column of A must be 191 r-l M ' (di e i +d2e2 H -drer) = dM 1ei+d2M *e2-t + d r M _ 1 ( = diM^bj, + d2M~1bJ2 + + drM~lbjr = diajr + d2aj2 + h drajv. The next theorem summarizes these results. Theorem 3.16. Let A be an m x n matrix of rank r, where r > 0, and let B be the reduced row echelon form of A. Then (a) The number of nonzero rows in B is r. (b) For each i = 1 ,2 , . . . , r, there is a column bji of B such that bjt = e .^ (c) The columns of A numbered ji,j2,---, jr nre linearly independent. (d) For each k = 1 ,2 , . . . n, if column k ofB is dei +d2e2 - -drer, then column k of A is dOj1 + d2aj2 + + dra,jr. Corollary. The reduced row echelon form of a matrix is unique. Proof. Exercise. (See Exercisel5.) M Example 2 Let / A = (2 4 6 2 4 1 2 3 1 1 2 4 8 0 0 3 6 7 5 9/ The reduced row echelon form of A is B = (I 0 0 ^ 2 0 0 0 0 1 0 0 4 - 1 0 0 0 0 1 0 / Since B has three nonzero rows, the rank of A is 3. The first, third, and fifth columns of B are ei,e2, and 63; so Theorem 3.16(c) asserts that the first, third, and fifth columns of A are linearly independent. Let the columns of A be denoted ai, a2, a$, 04, and a$. Because the second column of B is 2ei, it follows from Theorem 3.16(d) that a2 = 2a-i, as is easily checked. Moreover, since the fourth column of B is 4ei + (l)e2, the same result shows that 04 = 4a + (l)as. 192 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations In Example 6 of Section 1.6, we extracted a basis for R3 from the gener-ating set S = {(2, -3 ,5) , (8, -12,20), (1,0, -2) , (0,2, -1) , (7,2,0)}. The procedure described there can be streamlined by using Theorem 3.16. We begin by noting that if S were linearly independent, then S would be a basis for R3. In this case, it is clear that S is linearly dependent because S contains more than dim(R3) = 3 vectors. Nevertheless, it is instructive to consider the calculation that is needed to determine whether S is linearly dependent or linearly independent. Recall that S is linearly dependent if there are scalars Ci,C2,C3,C4, and C5, not all zero, such that c i (2 , -3 ,5)+C2(8 , -12 ,20)+c 3 ( l ,0 , -2)+c 4 (0 ,2 , - l )+ C 5 (7 ,2 ,0) = (0,0,0). Thus S is linearly dependent if and only if the system of linear equations 2ci + 8c2 + c3 + 7c5 = 0 -3ci - 12c2 + 2c4 + 2c5 = 0 5ci + 20c2 - 2c3 - c4 = 0 has a nonzero solution. The augmented matrix of this system of equations is A= - 3 -12 5 20 and its reduced row echelon form is B = Using the technique described earlier in this section, we can find nonzero solutions of the preceding system, confirming that 5 is linearly dependent. However, Theorem 3.16(c) gives us additional information. Since the first, third, and fourth columns of B are e,, e2, and 63, we conclude that the first, third, and fourth columns of A are linearly independent. But the columns of A other than the last column (which is the zero vector) are vectors in S. Hence 0 = {(2, -3 ,5 ) , (1,0, -2) , (0 ,2 , -1)} is a linearly independent subset of S. If follows from (b) of Corollary 2 to the replacement theorem (p. 47) that 0 is a basis for R3. Because every finite-dimensional vector space over F is isomorphic to Fn for some n, a similar approach can be used to reduce any finite generating set to a basis. This technique is illustrated in the next example. Sec. 3.4 Systems of Linear EquationsComputational Aspects 193 Example 3 The set S = {2+x+2x 2+3x 3 ,4+2x+4x 2+6x 3 ,6+3x+8x 2+7x 3 ,2+x+5x 3 ,4+x+9x 3} generates a subspace V of Ps(R). To find a subset of S that is a basis for V, we consider the subset Sf = {(2,1,2,3), (4,2,4,6), (6,3,8,7), (2,1,0,5), (4,1,0,9)} consisting of the images of the polynomials in S under the standard repre-sentation of Ps(R) with respect to the standard ordered basis. Note that the 4 x 5 matrix in which the columns are the vectors in S' is the matrix A in Example 2. From the reduced row echelon form of A, which is the matrix B in Example 2, we see that the first, third, and fifth columns of A are linearly independent and the second and fourth columns of A are linear combinations of the first, third, and fifth columns. Hence {(2,1,2,3), (6,3,8,7), (4,1,0,9)} is a basis for the subspace of R4 that is generated by S'. It follows that {2 + x + 2x2 + 3x3 ,6 + 3x + 8x2 + 7x3,4 + x + 9x3 } is a basis for the subspace V of Ps(i?.). We conclude this section by describing a method for extending a linearly independent subset S of a finite-dimensional vector space V to a basis for V. Recall that this is always possible by (c) of Corollary 2 to the replacement theorem (p. 47). Our approach is based on the replacement theorem and assumes that we can find an explicit basis 0 for V. Let S' be the ordered set consisting of the vectors in S followed by those in 0. Since 0 C S', the set S' generates V. We can then apply the technique described above to reduce this generating set to a basis for V containing S. Example 4 Let V = {(xi,X2,x3,X4,x5) e R5: xi +7x2 + 5x3 - 4x4 + 2x5 = 0}. It is easily verified that V is a subspace of R5 and that S = {(-2,0,0, -1 , -1) , (1,1, - 2 , - 1 , -1) , ( -5,1,0,1,1)} is a linearly independent subset of V. 194 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations To extend 5 to a basis for V, we first obtain a basis 0 for V. To do so, we solve the system of linear equations that defines V. Since in this case V is defined by a single equation, we need only write the equation as Xi = 7x2 5x3 + 4x4 2x5 and assign parametric values to X2, X3, X4, and X5. If X2 = 1, X3 = t2, X4 = 3, and X5 = 4, then the vectors in V have the form (xi,x2,X3,X4,x5) = (-7i -5t2+4t3 -2t4,ti,t2,t3,t4) = i i ( - 7 , l , 0 , 0 , 0 ) + ^ ( - 5 , 0 , l , 0 , 0 ) + 3 ( 4 , 0 , 0 , l , 0 ) + t4(-2 ,0 ,0 ,0 , l ) . Hence 0= {(-7,1,0,0,0), ( -5,0,1,0,0) , (4,0,0,1,0), ( -2 ,0 ,0,0,1)} is a basis for V by Theorem 3.15. The matrix whose columns consist of the vectors in S followed by those in 0 is (-2 0 0 - 1 ^ 1 1 - 2 - 1 - 1 - 5 1 0 1 1 - 7 1 0 0 0 - 5 0 1 0 0 4 0 0 1 0 -2 0 0 0 l) and its reduced row echelon form is / l 0 0 1 1 0 - 1 0 1 0 0 - . 5 0 0 0 0 1 1 .5 0 0 0 0 0 0 0 1 - 1 o 0 0 0 0 0 oy Thus { ( - 2 , 0 , 0 , - 1 , - 1 ) , ( 1 , 1 , - 2 , - 1 , - 1 ) , (-5,1,0,1,1) , (4,0,0,1,0)} is a basis for V containing S. EXERCISES 1. Label the following statements as true or false. (a) If (A'b') is obtained from (Ab) by a finite sequence of elementary column operations, then the systems Ax = b and ^4'x = b' are equivalent. Sec. 3.4 Systems of Linear EquationsComputational Aspects 195 (b) If (A'b') is obtained from (^4|6) by a finite sequence of elemen-tary row operations, then the systems ^4x = 6 and ^4'x = b' are equivalent. (c) If A is an n x n matrix with rank n, then the reduced row echelon form of A is In-(d) Any matrix can be put in reduced row echelon form by means of a finite sequence of elementary row operations. (e) If (Ab) is in reduced row echelon form, then the system Ax = 6 is consistent. (f) Let J4X = 6 be a system of m linear equations in n unknowns for which the augmented matrix is in reduced row echelon form. If this system is consistent, then the dimension of the solution set of Ax = 0 is n r, where r equals the number of nonzero rows in A. (g) If a matrix A is transformed by elementary row operations into a matrix A' in reduced row echelon form, then the number of nonzero rows in A' equals the rank of A. 2. Use Gaussian elimination to solve the following systems of linear equa-tions. (a) (c) (d) (e) (g) (h) Xi + 2X2 X3 = 1 2xi + 2x2 + X3 = 1 (b) 3xi + 5x2 2.X3 1 xi + 2x2 + 2x4 = 6 3xi + 5x2 X3 + 6x4 = 17 2xi + 4x2 + x3 + 2x4 = 12 2xi 7x3 + HX4 = 7 x x2 2x3 + 3x4 = 7 2xi X2 + 6x3 + 6x4 = 2 2X] + X2 4X3 ~' 3X4 = 0 3xi 2x2 + 9^ 3 + IOX4 = 5 Xi -2xi -3xi -Xi - 2x2 - X3 - 3x2 +- x3 - 5x2 + 5x3 = 1 = 6 = 7 = 9 xi - 4x2 - x3 + x4 = 3 2xi - 8x2 + X3 4x4 = 9 -X! + 4x2 - 2x3 + 5x4 = -6 2xi 2x2 X3 + 6x4 2x5 = 1 .X'i X2 + X3 + 2x4 X5 = 2 4xi - 4x2 + 5x3 + 7x4 - x5 = 6 3xi X2 + X3 X4 + 2x5 = 5 X] - x2 - x3 - 2x4 - x5 = 2 5xi - 2x2 + x-s - 3x4 + 3x5 = 10 2xi X2 2x4 + X5 = 5 xi + 2x2 X3 + 3x4 = 2 (f) 2xi + 4x2 - x3 + 6x4 = 5 x2 + 2x4 = 3 196 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (i) (j) 3xi - x2 + 2x3 + 4x4 + x5 = 2 Xi x2 + 2X3 + 3X4 + x5 = 1 2xi 3x2 + 6x3 + 9x4 + 4x5 = 5 7xi 2x2 + 4x3 + 8x4 + X5 = 6 2xi + 3x3 ~ 4x5 5 3xi 4x2 + 8x3 + 3x4 = 8 Xi x2 + 2x3 + X4 X5 = 2 2xi + 5x2 9x3 3x4 5x5 = 8 3. Suppose that the augmented matrix of a system .Ax = 6 is transformed into a matrix (A'b') in reduced row echelon form by a finite sequence of elementary row operations. (a) Prove that vank(A') ^ rank(,4/|6/) if and only if (^4'|6') contains a row in which the only nonzero entry lies in the last column. (b) Deduce that Ax = 6 is consistent if and only if (^4'|6') contains no row in which the only nonzero entry lies in the last column. 4. For each of the systems that follow, apply Exercise 3 to determine whether the system is consistent. If the system is consistent, find all solutions. Finally, find a basis for the solution set of the corresponding homogeneous system. X + X2 3x3 + X4 = 2 (b) xi + x2 + x3 - X4 = 2 xi + x2 - x 3 = 0 xi + 2x2 X3 + X4 = 2 (a) 2xi + x2 + x3 - x4 = 3 xi + 2x2 ~~ 3x3 + 2.T4 = 2 Xi + X2 3X3 + X4 = 1 (c) Xi + X2 + X3 - X4 = 2 Xi + X2 X3 = 0 5. Let the reduced row echelon form of A be ' 1 0 2 0 - 2 ' 0 1 - 5 0 - 3 J) 0 0 1 6, Determine A if the first, second, and fourth columns of A are and respectively. 6. Let the reduced row echelon form of A be / l - 3 0 4 0 0 0 0 0 0 0 5 1 3 0 0 0 1 0 0 0 2 - 1 0 / Sec. 3.4 Systems of Linear EquationsComputational Aspects Determine A if the first, third, and sixth columns of A are 197 / 1 - 2 - 1 , *) i >> 1 2 -4j and ( 3 - 9 2 V V respectively. 7. It can be shown that the vectors u = (2, 3,1), u2 = (1,4, 2), uz = (-8,12, -4) , w4 = (1,37, -17), and u5 = ( -3 , -5 ,8 ) generate R3. Find a subset of {u, u2,us, U4,u$} that is a basis for R3. 8. Let W denote the subspace of R5 consisting of all vectors having coor-dinates that sum to zero. The vectors i = (2 , -3 ,4 , -5 ,2 ) , 3 = (3 , -2 ,7 , -9 ,1 ) , u5 = ( -1 ,1 ,2 ,1 , -3 ) , u7 = ( 1 ,0 , -2 ,3 , -2 ) , and u2 = ( - 6 , 9 , - 1 2 , 1 5 , - 6 ) , u4 = (2 , -8 ,2 , -2 ,6 ) , u6 = (0 , -3 , -18 ,9 ,12) , u8 = ( 2 , - l , l , - 9 , 7 ) generate W. Find a subset of {ui,u2,..., ug} that is a basis for W. 9. Let W be the subspace of M2x2(R) consisting of the symmetric 2 x 2 matrices. The set S = 1 2 2 3 2 1 1 9 1 - 2 -2 4 -1 2 2 - 1 generates W. Find a subset of S that is a basis for W. 10. Let V = {(xi,x2,X3,X4,x5) G R5: Xi - 2x2 + 3x3 - x4 + 2x5 = 0}. (a) Show that S = {(0,1,1,1,0)} is a linearly independent subset of V. (b) Extend S to a basis for V. 11. Let V be as in Exercise 10. (a) Show that S = {(1,2,1,0,0)} is a linearly independent subset of V. (b) Extend S to a basis for V. 12. Let V denote the set of all solutions to the system of linear equations ^i #2 + 2x4 3x5 + XQ = 0 2xi X2 X3 + 3x4 4x5 + 4x6 = 0-198 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (a) Show that S = {(0, -1,0,1,1,0) , (1,0,1,1,1,0)} is a linearly inde-pendent subset of V. (b) Extend 5 to a basis for V. 13. Let V be as in Exercise 12. (a) Show that S = {(1,0,1,1,1,0), (0,2,1,1,0,0)} is a linearly inde-pendent subset of V. (b) Extend S to a basis for V. 14. If (Ab) is in reduced row echelon form, prove that A is also in reduced row echelon form. 15. Prove the corollary to Theorem 3.16: The reduced row echelon form of a matrix is unique. INDEX OF DEFINITIONS FOR CHAPTER 3 Augmented matrix 161 Augmented matrix of a system of lin-ear equations 174 Backward pass 186 Closed model of a sirnpic economy 176 Coefficient matrix of a system of lin-ear equations 169 Consistent system of linear equations 169 Elementary column operation 148 Elementary matrix 149 Elementary operation 148 Elementary row operation 148 Equilibrium condition for a simple economy 177 Equivalent systems of linear equa-tions 182 Forward pass 186 Gaussian elimination 186 General solution of a system of linear equations 189 Homogeneous system correspond-ing to a nonhomogeneous system 172 Homogeneous system of linear equa-tions 171 Inconsistent system of linear equa-tions 169 Input- output matrix 177 Nonhomogeneous system of linear equations 171 Nonnegative vector 177 Open model of a simple economy 178 Positive matrix 177 Rank of a matrix 152 Reduced row echelon form of a ma-trix 185 Solution to a system of linear equa-tions 169 Solution set of a system of equations 169 System of linear equations 169 Type 1, 2, and 3 elementary opera-tions 148 4 D e t e r m i n a n t s 4.1 4.2 4.3 4.4 4.5* Determinants of Order 2 Determinants of Order n Properties of Determinants Summary Important Facts about Determinants A Characterization of the Determinant JL he determinant, which has played a prominent role in the theory of lin-ear algebra, is a special scalar-valued function defined on the set of square matrices. Although it still has a place in the study of linear algebra and its applications, its role is less central than in former times. Yet no linear algebra book would be complete without a systematic treatment of the determinant, and we present one here. However, the main use of determinants in this book is to compute and establish the properties of eigenvalues, which we discuss in Chapter 5. Although the determinant is not a linear transformation on M n x n ( F ) for n > 1, it does possess a kind of linearity (called n-linearity) as well as other properties that are examined in this chapter. In Section 4.1, we consider the determinant on the set of 2 x 2 matrices and derive its important properties and develop an efficient computational procedure. To illustrate the important role that determinants play in geometry, we also include optional material that explores the applications of the determinant to the study of area and orientation. In Sections 4.2 and 4.3, we extend the definition of the determinant to all square matrices and derive its important properties and develop an efficient computational procedure. For the reader who prefers to treat determinants lightly, Section 4.4 contains the essential properties that are needed in later chapters. Finally, Section 4.5, which is optional, offers an axiomatic approach to determinants by showing how to characterize the determinant in terms of three key properties. 4.1 DETERMINANTS OF ORDER 2 In this section, we define the determinant of a 2 x 2 matrix and investigate its geometric significance in terms of area and orientation. 199 200 Chap. 4 Determinants Definition. If A = a b c d is a 2 x 2 matrix with entries from a field F, then we define the determinant of A, denoted det(^4) or A, to be the scalar ad be. Example 1 For the matrices A = 1 2 and B = 3 2 3 4 ; ~ Q 4 in M2x2(R), we have det(A) = 1 - 4 - 2 - 3 = - 2 and det(B) = 3 - 4 - 2 - 6 = 0. For the matrices A and B in Example 1, we have '4 4N A + B = 9 8 and so / det(A + B) = 4-8 - 4-9 = - 4 . Since det ( 4 + B) ^ det(A) + det(B), the function det: M2X2(-R) * i? is not a linear transformation. Nevertheless, the determinant does possess an important linearity property, which is explained in the following theorem. Theorem 4.1. The function det: M2x2(-F') > F is a linear function of each row of a 2 x 2 matrix when the other row is held fixed. That is, ifu, v, and w are in F2 and k is a scalar, then and ^ l U + kv)=det(' w w w A; det w det I I'. )=dct[W)+kdetln U + KV J U I V Proof. Let u = (ai,a2), v (b,b2), and w = (c,c2) be in F2 and A; be a scalar. Then det ID kdctiV)=det(ai ^ WI Ci c2 A-d,. { h h C] c2 Sec. 4.1 Determinants of Order 2 201 = {a(-2 - &2C1) + k{bC2 - b>202 Chap. 4 Determinants Therefore det(A) = AnA22 - A !2A2i ^ 0. On the other hand, if A2i ^ 0, we see that det(A) ^ 0 by adding - A n / A 2 i times row 2 of A to row 1 and applying a similar argument. Thus, in either case, det(A) ^ 0. In Sections 4.2 and 4.3, we extend the definition of the determinant to nxn matrices and show that Theorem 4.2 remains true in this more general context. In the remainder of this section, which can be omitted if desired, we explore the geometric significance of the determinant of a 2 x 2 matrix. In particular, we show the importance of the sign of the determinant in the study of orientation. The Area of a Parallelogram By the angle between two vectors in R2, we mean the angle with measure 9 (0 < 9 < IT) that is formed by the vectors having the same magnitude and direction as the given vectors but emanating from the origin. (See Figure 4.1.) Figure 4.1: Angle between two vectors in R' If 0 = {u,v} is an ordered basis for R2, we define the orientation of 0 to be the real number O del del (The denominator of this fraction is nonzero by Theorem 4.2.) Clearly 0 1*' I I. v Notice that ei O ( w = 1 and 0 1 ~l 1 -1 . e 2 / V-C2. ' I Recall that a coordinate system {u, v} is called right-handed if u can be rotated in a counterclockwise direction through an angle 9 (0 < 9 < n) Sec. 4.1 Determinants of Order 2 203 to coincide with v. Otherwise {u, v} is called a left-handed system. (See Figure 4.2.) In general (see Exercise 12), V I V A right-handed coordinate system A left-handed coordinate system Figure 4.2 O = 1 if and only if the ordered basis {u, v} forms a right-handed coordinate system. For convenience, we also define () = 1 if {u.v} is linearly dependent. Any ordered set {u, v} in R2 determines a parallelogram in the following manner. Regarding u and v as arrows emanating from the origin of R2, we call the parallelogram having u and v as adjacent sides the parallelogram determined by u and v. (See Figure 4.3.) Observe that if the set {//, v) A V A ^ M ' x Figure 4.3: Parallelograms determined by u and v is linearly dependent (i.e.. if u and r are parallel), then the 'parallelogram' determined by u and v is actually a line segment, which we consider to be a degenerate parallelogram having area zero. 204 Chap. 4 Determinants There is an interesting relationship between A the area of the parallelogram determined by u and v, and det which we now investigate. Observe first, however, that since det may be negative, we cannot expect that < - * * : But we can prove that A | W = 0 ' l - d e t v v I W u from which it follows that Our argument that det > : - : - : employs a technique that, although somewhat indirect, can be generalized to Rn. First, since 0 [l) = H , we may multiply both sides of the desired equation by to obtain the equivalent form ' : * : - * : Sec. 4.1 Determinants of Order 2 205 We establish this equation by verifying that the three conditions of Exercise 11 are satisfied bv the function < = : - A : (a) We begin by showing that for any real number c l ' ) - e . * ( ' cv J v Observe that this equation is valid if c = 0 because ' J = l o - A o 1 So assume that c ^ 0. Regarding cv as the base of the parallelogram deter-mined by u and cv, we see that A [ J = base x altitude = |c|(length of v)(altitude) = c A ( ) . since the altitude h of the parallelogram determined by u and cv is the same as that in the parallelogram determined by u and v. (See Figure 4.4.) Hence Figure 4.4 ' : = : - M ; : c v^ |c|-A - o r : -A : )=c.sr A similar argument shows that Si0') ~e.t[u v v 206 Chap. 4 Determinants We next prove that 1, ) = * ( ' mi + bw J w for any u, w R2 and any real numbers a and b. Because the parallelograms determined by u and w and by u and u + w have a common base u and the same altitude (see Figure 4.5), it follows that If a = 0. then Figure 4.5 = A u + w = 61 ')=b-of au + bw J bw I w by the first paragraph of (a). Otherwise, if a ^ 0, then 81 u ) =a-6 'i, ) =a-6 [ /,' au + bwj u +-ic I -w So the desired conclusion is obtained in either case. We are now able to show that s( ' )=s(u) + 4 ' Vl+V2j V'/ V'-' for all u,v,V2 G R2. Since the result is immediate if u = 0, we assume that u / 0. Choose any vector w R2 such that {(/.//} is linearly independent. Then for any vectors V[,t'2 R2 there exist scalars a, and b, such that Vi = a,;W + bite (i = 1, 2). Thus si ; =8 vi +v2 II [O] + 0,2)11 + (6i + b2 = (bl+b2)6 ir Sec. 4.1 Determinants of Order 2 1, )+*( ! * = * ( ' au + biwl a2u--b2wl V A similar argument shows that 51U1+U2)=S(U1)+5(U2 for all ui,U2,v 6 R2-(b) Since A f = 0, it follows that 6 C^ = 0 f A (^) = 0 for any u G R2. (c) Because the parallelogram determined by ei and e2 is the unit square, ' j '2 e2 Therefore 5 satisfies the three conditions of Exercise 11, and hence 5 = det. So the area of the parallelogram determined by u and v equals r . d e t r v I v Thus we see, for example, that the area of the parallelogram determined by u = (-1,5) and v = (4, -2 ) is det det 4 - 2 = 18. EXERCISES 1. Label the following statements as true or false. (a) The function det: W2x2(F) * F is a linear transformation. (b) The determinant of a 2 x 2 matrix is a linear function of each row of the matrix when the other row is held fixed. (c) If A e M2x2{F) and det(,4) = 0, then A is invertible. (d) If u and v are vectors in R2 emanating from the origin, then the area of the parallelogram having u and v as adjacent sides is det 208 Chap. 4 Determinants (e) A coordinate system is right-handed if and only if its orientation equals 1. 2. Compute the determinants of the following matrices in M2x2(-ft)-'I 1 ) ( 1 i ) W (a - f (a) 3. Compute the determinants of the following matrices in M2x2(C). ' - 1 + i 1 - 4 A . . . / 5 - 2 i 6 + 4A , v /2i 3 3 + 2* 2 - 3 i ] ( b ) ( - 3 + i 7* ( C ) I 4 6i (a) 4. For each of the following pairs of vectors u and v in R , compute the area of the parallelogram determined by u and v. (a) u = (3, - 2 ) and v = (2, 5) (b) u = (1,3) andu = (-3,1) (c) u = ( 4 , - l ) a n d v = ( - 6 , - 2 ) (d) u = (3,4) and v = (2, -6 ) 5. Prove that if B is the matrix obtained by interchanging the rows of a 2 x 2 matrix A, then det(B) = - det(A). 6. Prove that if the two columns of A G M2x2(F) are identical, then dct(A) = 0. 7. Prove that det(A') = det(.A) for any A 6 M2X2(-F)-8. Prove that if A e M2X2(^) is upper triangular, then det(^l) equals the product of the diagonal entries of A. 9. Prove that det{AB) = det(A) det(B) for any A, Be M2X2{F). 10. The classical adjoint of a 2 x 2 matrix A G M2X2(-F) is the matrix C = Aoo -A 12 -A: 21 A Prove that (a) 0 4 = AC = [det(A)]7. (b) det(C) = det(A). (c) The classical adjoint of A1 is C*. (d) If A is invertible, then A'1 = [ d e t ^ ^ C . 11. Let 8: N2x2{F) F be a function with the following three properties. (i) 8 is a linear function of each row of the matrix when the other row is held fixed. (ii) If the two rows of A E M2x2(F) are identical, then 8(A) = 0. Sec. 4.2 Determinants of Order n (iii) li I is the 2 x 2 identity matrix, then 8(1) = 1. Prove that 8(A) = det(A) for all A G M2X2(F). (This result is general-ized in Section 4.5.) 12. Let {u.r} be an ordered basis for R2. Prove that 0 -if and only if {u, v} forms a right-handed coordinate system. Hint: Recall the definition of a rotation given in Example 2 of Section 2.1. 4.2 DETERMINANTS OF ORDER 71 In this section, we extend the definition of the determinant to n x n matrices for n > 3. For this definition, it is convenient to introduce the following notation: Given A G M n X n(F) , for n > 2, denote the (n - 1) x (n - 1) matrix obtained from A by deleting row i and column j by Aij. Thus for / . 2 3 A=4 5 6 eM3x3(R), 7 8 9 / we have A u id for 5 6 8 9 B = Aia = I 5 7 8 and A-12 = 1 3 4 6 / 1 - 3 2 V-2 - 1 2 4 1 - 5 - 3 6 - 4 1 - 1 8 1 G M4x4(tf), we have Bo* = i - l - r 2 - 5 8 | and 4 ; -2 6 1 Definitions. Let A G MnXn(F). 7f n = 1. so that A = (An), we define det(A) An- For n > 2, we define det(A) recursively as n det(A) = j ^ ( - l ) 1 + i A y d e t ( i y ) . 210 Chap. 4 Determinants The scalar det (A) is called the determinant of A and is also denoted byA. The scalar (-l)i+j det(A{j) is called the co factor of the entry of A in row i, column j. Letting Cij = ( - l ) ^ d e t ( A V ) denote the cofactor of the row i, column j entry of A, we can express the formula for the determinant of A as det(A) = A u d i + Auci2 + + Aici. Thus the determinant of A equals the sum of the products of each entry in row 1 of A multiplied by its cofactor. This formula is called cofactor expansion along the first row of A. Note that, for 2 x 2 matrices, this definition of the determinant of A agrees with the one given in Section 4.1 because det(A) = Au(-l)l+1 det(An) + A 1 2 ( - l ) 1 + 2 det (A 1 2 ) = AnA 2 2 - A12A21. Example 1 Let 1 3 - 3 - 3 - 5 2 M3x3(i?). - 4 4 - 6 / Using cofactor expansion along the first row of A, we obtain det(A) = ( - l ) 1 + 1 A n - det(An) + ( - l ) 1 + 2 A i 2 - det(A12) + ( - l ) 1 + 3 A 1 3 -de t (A 1 3 ) ,2'- .Sec. 4.2 Determinants of Order // 211 Example 2 Let / 0 1 3 B= - 2 - 3 - 5 GM3x3(i?). 4 - 4 4 / Using cofactor expansion along the first row of B, we obtain det(B) = (-l)1 + lBu det(Bn) + (-l)1+2Bi2- det(BV2) + (-iy+3BV3.det(B13) = ( - l ) 2 ( 0 ) - d e t ( ^ - ^ + ( - l ) 3 ( l ) . d e t ( - ^ ~ + ( - l ) 4 (3 ) -de t - 2 - 3 4 - 4 , = 0 - 1 [-2(4) - (-5)(4)] + 3 [ -2 ( -4) - (-3)(4)] = 0 - 1(12)+ 3(20) = 48. Example 3 Let C = ( 2 0 0 1 0 1 3 - 3 -2 - 3 - 5 2 4 - 4 4 -ey GM4 x 4(f i ) . Using cofactor expansion along the first row of C and the results of Examples 1 and 2, we obtain det(C) = (-1)2(2) . det(Cn) + (-1)3(0)- det(C12) + (-1)4(0) det(C13) + (-1)5(1) . det(C14) / 1 3 - 3 = ( - l ) 2 (2) -de t - 3 - 5 2 + 0 + 0 - 4 4 - 6 / 0 1 f ( - l ) 5 ( l ) . d e t I - 2 - 3 - 5 4 - 4 = 2(40)+ 0 + 0 - 1(48) = 32. 212 Chap. 4 Determinants Example 4 The determinant of the n x n identity matrix is 1. We prove this assertion by mathematical induction on n. The result is clearly true for the l x l identity matrix. Assume that the determinant of the (n 1) x (n 1) identity matrix is 1 for some n > 2, and let I denote the nxn identity matrix. Using cofactor expansion along the first row of J, we obtain det(I) = (-1)2(1) det(Jn) + (-1)3(0) det(/12) + + ( - l ) l + n ( 0 ) . d e t ( / l n ) = l ( l ) + 0 + --- + 0 = 1 because In is the (n - 1) x (n - 1) identity matrix. This shows that the determinant of the nxn identity matrix is 1, and so the determinant of any identity matrix is 1 by the principle of mathematical induction. As is illustrated in Example 3, the calculation of a determinant using the recursive definition is extremely tedious, even for matrices as small as 4 x 4 . Later in this section, we present a more efficient method for evaluating determinants, but we must first learn more about them. Recall from Theorem 4.1 (p. 200) that, although the determinant of a 2 x 2 matrix is not a linear transformation, it is a linear function of each row when the other row is held fixed. We now show that a similar property is true for determinants of any size. Theorem 4.3. The determinant of an n x n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 < r < n, we have det 'I o r_i u + kv a r+i = det ( i a r_ i u O-r+l + fcdet ( a, a r _! v ar+i whenever k is a scalar and u, v, and each a^ are row vectors in Fn. Proof. The proof is by mathematical induction on n. The result is imme-diate if n = 1. Assume that for some integer n > 2 the determinant of any (n l ) x ( n 1) matrix is a linear function of each row when the remaining Sec. 4.2 Determinants of Order n 213 rows are held fixed. Let A be an n x n matrix with rows a, a2,..., a n , respec-tively, and suppose that for some r (1 < r < n), we have ar = u + kv for some u, v G Fn and some scalar k. Let u = (&i, b2,..., bn) and v = (c, c2, , cn), and let and C be the matrices obtained from A by replacing row r of A by u and v, respectively. We must prove that det(A) = det(B) + fcdet(C). We leave the proof of this fact to the reader for the case r = 1. For r > 1 and 1 ^ j < n-, the rows of Aij, JBIJ, and Cy are the same except for row r 1. Moreover, row r 1 of Au is (bi+kci,..., bj^i + kCj-i,bj+i + fccj+i,..., 6 + fccn), which is the sum of row r 1 of Bj and fc times row r 1 of Cij. Since Z?ij and Cij are (n 1) x (n 1) matrices, we have det(Aij) = det(Z?ij) + kdet(Cij) by the induction hypothesis. Thus since Aj = Z?ij = Cj, we have n det(A) = ^2(-l)1+j Au de t ( i i j ) = y > i ) ' + ' A - det(fiy) +Ardet(Cij - ( - l ) 1 + i A i i dct(Bl3) + * 2 ( - l ) 1 + i A y * det(Cy) = det() + fcdet(C). This shows that the theorem is true for nxn matrices, and so the theorem is true for all square matrices by mathematical induction. Corollary. If A & M n x n (F ) has a row consisting entirely of zeros, then det(A) = 0. Proof. See Exercise 24. The definition of a determinant requires that the determinant of a matrix be evaluated by cofactor expansion along the first row. Our next theorem shows that the determinant of a square matrix can be evaluated by cofactor expansion along any row. Its proof requires the following technical result. Lemma. Let B G M n x n (F) , where n > 2. If row i of B equals e^ for some k (1 < k < n), then det(B) = (-l)i+k det(Bik). 214 Chap. 4 Determinants Proof. The proof is by mathematical induction on n. The lemma is easily proved for n = 2. Assume that for some integer n > 3, the lemma is true for (n 1) x (n 1) matrices, and let B be an n x n matrix in which row i of B equals e* for some /c (1 < k < n). The result follows immediately from the definition of the determinant if i = 1. Suppose therefore that 1 < i < n. For each j ^ A: (1 < j < n), let Cij denote the (n 2) x (n 2) matrix obtained from B by deleting rows 1 and i and columns j and k. For each j , row i 1 of Z?ij is the following vector in F n _ 1 : efc_i i f j< fc 0 if j = A; efc if j > k. Hence by the induction hypothesis and the corollary to Theorem 4.3, we have ( - l ) - i )+ (* - i )de t (Cy if j k. fV< Therefore n det(B) = ^ ( - l ) 1 + ^ l j - d e t ( B l i ; j=i = ^ ( - l ) 1 + ^ i r det(By) + ( - l ) 1 + ^ det(Bn; jfc = ^ ( - l ^ Z V [(_i)(*-D+(*-i) det(C^) 3 B y . [ ( - ! ) (* - ! )+*de t (^ ) j>fc = ( - 1 .i+fc ^ ( - l )1 +^ i r d . e t ( C i jfc Because the expression inside the preceding bracket is the cofactor expan-sion of Bik along the first row, it follows that (let() = (-l) i+ fcdet(G i f c)-This shows that the lemma is true for n x n matrices, and so the lemma is true for all square matrices by mathematical induction. Sec. 4.2 Determinants of Order n 215 We are now able to prove that cofactor expansion along any row can be used to evaluate the determinant of a square matrix. Theorem 4.4. The determinant of a square matrix can be evaluated by cofactor expansion along any row. That is, if A G M n x n (F ) , then for any integer i (I < i < n), n det(A) = ^ ( - l ) J + M ^ . d e t ( A ^ ) . i - i Proof. Cofactor expansion along the first row of A gives the determinant of A by definition. So the result is true if i = 1. Fix i > 1. Row i of A can be written as X)?=i ^ijej- For 1 < j < n, let Bj denote the matrix obtained from A by replacing row i of A by ej. Then by Theorem 4.3 and the lemma, we have n n det(A) =^Aij det(Bj) = ^(-1)*+ JA;.,- det(A^). I 3=1 3=1 Corollary. If A G MnXn(F) has two identical rows, then det(A) = 0. Proof. The proof is by mathematical induction on n. We leave the proof of the result to the reader in the case that n = 2. Assume that for some integer n > 3, it is true for (n 1) x (n 1) matrices, and let rows r and s of A G M n x n (F ) be identical for r ^ s. Because n > 3, we can choose an integer i (1 < i < n) other than r and s. Now det(A) = ^ ( - l ) i + J A i r d e t ( A i 3=1 by Theorem 4.4. Since each Ay is an (n 1) x (n 1) matrix with two identical rows, the induction hypothesis implies that each det(A^) = 0, and hence det(A) = 0. This completes the proof for n x n matrices, and so the lemma is true for all square matrices by mathematical induction. It is possible to evaluate determinants more efficiently by combining co-factor expansion with the use of elementary row operations. Before such a process can be developed, we need to learn what happens to the determinant of a matrix if we perform an elementary row operation on that matrix. The-orem 4.3 provides this information for elementary row operations of type 2 (those in which a row is multiplied by a nonzero scalar). Next we turn our attention to elementary row operations of type 1 (those in which two rows are interchanged). 216 Chap. 4 Determinants Theorem 4.5. If A G Mn x n(F) and B is a matrix obtained from A by interchanging any two rows of A, then det(-B) = det(A). Proof. Let the rows of A G M n X n (F) be a, a2,..., a n , and let B be the matrix obtained from A by interchanging rows r and s, where r < s. Thus /ai A = a, and B = (ttl as ar anJ anJ Consider the matrix obtained from A by replacing rows r and s by ar + at By the corollary to Theorem 4.4 and Theorem 4.3, we have 0 = det ( ai ar + as ar + a. = det ( *i ar + a4 / a , + det ar + a. V an / On / an / fai / a i / a i / a A = det a r ar-det '/ det a n / Van / = 0 + det(A) + det(B) + 0. Therefore det(B) = -det(A). a. ar a n / det a n / We now complete our investigation of how an elementary row operation affects the determinant of a matrix by showing that elementary row operations of type 3 do not change the determinant of a matrix. Theorem 4.6. Let A G MnXn(F), and let B be a matrix obtained by adding a multiple of one row of A to another row of A. Then det(B) = det(A). L T-ZZ Sec. 4.2 Determinants of Order 217 Proof. Suppose that B is the nxn matrix obtained from A by adding k times row r to row s, where r ^ s. Let the rows of A be a, a2,..., an, and the rows of B be b,b2,..., bn. Then bi = ai for i ^ s and bs = as + kar. Let C be the matrix obtained from A by replacing row s with a r . Applying Theorem 4.3 to row s of B, we obtain det(B) = det(A) + Ardet(C) = det(A) because det(C) = 0 by the corollary to Theorem 4.4. In Theorem 4.2 (p. 201), we proved that a 2 x 2 matrix is invertible if and only if its determinant is nonzero. As a consequence of Theorem 4.6, we can prove half of the promised generalization of this result in the following corollary. The converse is proved in the corollary to Theorem 4.7. Corollary. If A M n x n ( F ) has rank less than n, then det(A) = 0. Proof. If the rank of A is less than n, then the rows ai, a2,..., an of A are linearly dependent. By Exercise 14 of Section 1.5, some row of A, say, row r, is a linear combination of the other rows. So there exist scalars Cj such that ar = ca H h Cr-icir-i + cr+iar+i -I h cnan. Let B be the matrix obtained from A by adding c? times row i to row r for each i ^ r. Then row r of B consists entirely of zeros, and so det(B) = 0. But by Theorem 4.6, det(B) = det(A). Hence det(A) = 0 . I The following rules summarize the effect of an elementary row operation on the determinant of a matrix A G M n X n(F) . (a) If B is a matrix obtained by interchanging any two rows of A, then det(B) = -det(A). (b) If B is a matrix obtained by multiplying a row of A by a nonzero scalar k, then det(B) = fcdet(A). (c) If B is a matrix obtained by adding a multiple of one row of A to another row of A, then det(B) = det (A). These facts can be used to simplify the evaluation of a determinant. Con-sider, for instance, the matrix in Example 1: -3 4 - 6 / Adding 3 times row 1 of A to row 2 and 4 times row 1 to row 3, we obtain M = 218 Chap. 4 Determinants Since M was obtained by performing two type 3 elementary row operations on A, we have det(A) = det(M). The cofactor expansion of M along the first row gives det(M) = ( -1) 1 + 1 (1) . det(Mn) + ( - l ) 1 + 2 (3)* det(M12) + ( - l ) 1 + 3 ( -3)*de t (M 1 3 ) . Both Mi2 and M3 have a column consisting entirely of zeros, and so det(Mi2) = det(Mi3) = 0 by the corollary to Theorem 4.6. Hence det(M) = ( -1) 1 + 1 (1) . det(Mn) = ( - 1 ) ^ ( 1 ) . det ( / 6 _~l = l [ 4 ( - 1 8 ) - ( - 7 ) ( 1 6 ) ] = 4 0 . Thus with the use of two elementary row operations of type 3, we have reduced the computation of det (A) to the evaluation of one determinant of a 2 x 2 matrix. But we can do even better. If we add 4 times row 2 of M to row 3 (another elementary row operation of type 3), we obtain P = Evaluating det(P) by cofactor expansion along the first row, we have det(P) = ( - l ) 1 + 1 ( l ) . d e t ( P n ) = ( - l ) 1 + 1 ( l ) - d e t ^ ~ ) =1*4.10 = 40, as described earlier. Since det(A) = det(M) = det(P), it follows that det(A) = 40. The preceding calculation of det(P) illustrates an important general fact. The determinant of an upper triangular matrix is the product of its diagonal entries. (See Exercise 23.) By using elementary row operations of types 1 and 3 only, we can transform any square matrix into an upper triangular matrix, and so we can easily evaluate the determinant of any square matrix. The next two examples illustrate this technique. Example 5 To evaluate the determinant of the matrix Sec. 4.2 Determinants of Order n 219 in Example 2, we must begin with a row interchange. Interchanging rows 1 and 2 of B produces C = By means of a sequence of elementary row operations of type 3, we can transform C into an upper triangular matrix: - 2 0 4 - 3 1 - 4 - 5 3 4 Thus det(C) = -2-1-24 = -48. Since C was obtained from B by an inter-change of rows, it follows that det(B) = - det(C) = 48. Example 6 The technique in Example 5 can be used to evaluate the determinant of the matrix C = 2 0 - 2 4 0 1 - 3 - 4 0 3 - 5 4 A - 3 2 - v in Example 3. This matrix can be transformed into an upper triangular matrix by means of the following sequence of elementary row operations of type 3: / 2 0 0 1 - 2 - 3 4 - 4 T hus det(C) 0 3 - 5 4 = 2-A - 3 2 -6) 1.4. (2 0 0 /2 0 0 4 = 32 0 1 - 3 - 4 0 1 0 0 0 3 - 5 4 0 3 4 0 A - 3 3 * f 2 0 0 - 8 / 0 A - 3 - 6 V 0 0 1 3 0 4 0 16 A - 3 - 6 -20^ Using elementary row operations to evaluate the determinant of a matrix, as illustrated in Example 6, is far more efficient than using cofactor expansion. Consider first the evaluation of a 2 x 2 matrix. Since det I , 1 = ad be, 220 Chap. 4 Determinants the evaluation of the determinant of a 2 x 2 matrix requires 2 multiplications (and 1 subtraction). For n > 3, evaluating the determinant of an n x n matrix by cofactor expansion along any row expresses the determinant as a sum of n products involving determinants of (n 1) x (n 1) matrices. Thus in all. the evaluation of the determinant of an n x n matrix by cofactor expansion along any row requires over n! multiplications, whereas evaluating the determinant of an n x n matrix by elementary row operations as in Examples 5 and 6 can be shown to require only (n3 + 2n 3)/3 multiplications. To evaluate the determinant of a 20 x 20 matrix, which is not large by present standards, cofactor expansion along a row requires over 20! ~ 2.4 x 1018 multiplica-tions. Thus it would take a computer performing one billion multiplications per second over 77 years to evaluate the determinant of a 20 x 20 matrix by this method. By contrast, the method using elementary row operations re-quires only 2679 multiplications for this calculation and would take the same computer less than three-millionths of a second! It is easy to see why most computer programs for evaluating the determinant of an arbitrary matrix do not use cofactor expansion. In this section, we have defined the determinant of a square matrix in terms of cofactor expansion along the first row. We then showed that the determinant of a square matrix can be evaluated using cofactor expansion along any row. In addition, we showed that the determinant possesses a number of special properties, including properties that enable us to calculate det(B) from det (A) whenever B is a matrix obtained from A by means of an elementary row operation. These properties enable us to evaluate determi-nants much more efficiently. In the next section, we continue this approach to discover additional properties of determinants. EXERCISES 1. Label the following statements as true or false. (a) The function det: M n X n (F) > F is a linear transformation. (b) The determinant of a square matrix can be evaluated by cofactor expansion along any row. (c) If two rows of a square matrix A are identical, then det(A) = 0. (d) If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = det(A). (e) If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A). (f) If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = kdct(A). (g) If A G M n x (F) has rank n, then det(A) = 0. (h) The determinant of an upper triangular matrix equals the product of its diagonal entries. Sec. 4.2 Determinants of Order n 221 2. Find the value of k that satisfies the following equation: /3a i 3a2 3a3 fa a2 a3 det I 36i 3o2 363 = kdct I &i b2 b3 . 3ci 3c2 3c3 / d e2 c3) 3. Find the value of k that satisfies the following equation: / 2a i 2a2 2a3 / o j a2 a 3 det 36i + 5d 362 + 5c2 363 + 5c3 = fcdct [ 6X b2 b3 . 7cj 7c2 7c3 / c c2 c 3 / 4. Find the value of k that satisfies the following equation: /bi+ci b2 + c2 b3 + c3 / o j a2 a3 ' det j ai + ci a2 + c2 a3 + c3 j = /edet I 6i 62 63 a , +61 a 2 + 6 2 a3 + 6 3 / d c2 c3/ In Exercises 5-12, evaluate the determinant of the given matrix by cofactor expansion along the indicated row. 5. 7. !). . ; : J V 2 3 0 / along the first row 0 I 2N -1 0 - 3 2 3 0, along the second row 0 1 + i 2 -2 i 0 1 r 3 4i 0 y along the third row (i. 8. 10. 1 0 2 0 1 5 - 1 3 0 / along the first row 1 0 2 0 1 5 ^-1 3 0 / along the third row (% 2 + i 0 - 1 3 2i 0 - 1 1 - ; / along the second row / 0 2 1 3 1 0 - 2 2 3 - 1 0 1 v - l 1 2 Oy along the fourth row 12. / 1 - 1 2 - A - 3 4 1 - 1 2 - 5 - 3 8 ^-2 6 - 4 I, along the fourth row 11. In Exercises 13-22, evaluate the determinant of the given matrix by any le-gitimate method. 222 Chap. 4 Determinants 21. 23. Prove that the determinant of an upper triangular matrix is the product of its diagonal entries. 24. Prove the corollary to Theorem 4.3. 25. Prove that det(ArA) = kn det(A) for any A G M n x n (F) . 26. Let A G M n x n (F ) . Under what conditions is det(-A) = det(A)? 27. Prove that if A G M n X n (F) has two identical columns, then det(A) = 0. 28. Compute det(Fj) if F,; is an elementary matrix of type i. 29. ' Prove that if E is an elementary matrix, then det(F') = det(F). 30. Let the rows of A G M n x n (F ) be a, a2,..., an, and let B be the matrix in which the rows are a n , a n _ i , . . . ,aj. Calculate det(Z?) in terms of det(A). 4.3 PROPERTIES OF DETERMINANTS In Theorem 3.1, we saw that performing an elementary row operation on a matrix can be accomplished by multiplying the matrix by an elementary matrix. This result is very useful in studying the effects on the determinant of applying a sequence of elementary row operations. Because the determinant Sec. 4.3 Properties of Determinants 223 of the nxn identity matrix is 1 (see Example 4 in Section 4.2), we can interpret the statements on page 217 as the following facts about the determinants of elementary matrices. (a) If E is an elementary matrix obtained by interchanging any two rows of/ , thendet(F) = -1 . (b) If E is an elementary matrix obtained by multiplying some row of / by the nonzero scalar k, then det(F) = k. (c) If E is an elementary matrix obtained by adding a multiple of some row of J to another row, then det(F) = 1. We now apply these facts about determinants of elementary matrices to prove that the determinant is a multiplicative function. Theorem 4.7. For any A, B G ,,(F), det(AP) = det(A)- det(P). Proof. We begin by establishing the result when A is an elementary matrix. If A is an elementary matrix obtained by interchanging two rows of / , then det(A) = -1 . But by Theorem 3.1 (p. 149), AB is a matrix obtained by interchanging two rows of B. Hence by Theorem 4.5 (p. 216), det(AP) = dct(B) = det(A)- det(B). Similar arguments establish the result when A is an elementary matrix of type 2 or type 3. (See Exercise 18.) If A is an nxn matrix with rank less than n, then det(A) = 0 by the corollary to Theorem 4.6 (p. 216). Since rank(AP>) < rank(A) < n by Theo-rem 3.7 (p. 159), we have det(AB) = 0. Thus det(AB) = det(A)- det(P) in this case. On the other hand, if A has rank n, then A is invertible and hence is the product of elementary matrices (Corollary 3 to Theorem 3.6 p. 159), say, A = Em E2Ei. The first paragraph of this proof shows that det(AB) = det(F r = det(F r E2ElB) d e t ( F m _ ! - - - F 2 F i ) = det(Fm) det(F2) det(Fi) det(P) = de t (F m - - -F 2 Fi ) -de t (P ) = det(A)-det(P). I Corollary. A matrix A G M n x n (F ) is invertible if and only if det(A) 7^ 0. Furthermore, if A is invertible, then det(A_1) = -7-. v ; det(A) Proof. If A G M n X n (F) is not invertible, then the rank of A is less than n. So det(A) = 0 by the corollary to Theorem 4.6 (p, 217). On the other hand, if A G MnXTl(F) is invertible, then det(A). det(A_1) = det(AA_1) = det(/) = 1 224 Chap. 4 Determinants by Theorem 4.7. Hence det(A) ^ 0 and det(A~1) = det(A) In our discussion of determinants until now, we have used only the rows of a matrix. For example, the recursive definition of a determinant involved cofactor expansion along a row, and the more efficient method developed in Section 4.2 used elementary row operations. Our next result shows that the determinants of A and A1 are always equal. Since the rows of A are the columns of A6, this fact enables us to translate any statement about determi-nants that involves the rows of a matrix into a corresponding statement that involves its columns. Theorem 4.8. For any A G Mn x n(F), det(A') = det(A). Proof. If A is not invertible, then rank(A) < n. But rank(A') = rank(A) by Corollary 2 to Theorem 3.G (p. 158), and so A1 is not invertible. Thus det(A') = 0 = det(A) in this case. On the other hand, if A is invertible, then A is a product of elementary matrices, say A = Em- E2E. Since det(F,) = det(F|) for every i by Exercise 29 of Section 4.2, by Theorem 4.7 we have det(A') = det(EEl2 ...fit ) = det (Fi ) .det (F 2 ) = det (F0-= det (F m ) . = det(Fm = det(A). det(F2)---- det(F2) F 2 F 0 det(F^) Sec. 4.3 Properties of Determinants 225 where Mk is the nxn matrix obtained from A by replacing column k of A byb. Proof. If det (A) ^ 0, then the system Ax = b has a unique solution by the corollary to Theorem 4.7 and Theorem 3.10 (p. 174). For each integer k (1 < k < n), let ak denote the kth column of A and Xk denote the matrix obtained from the nxn identity matrix by replacing column k by x. Then by Theorem 2.13 (p. 90), AXk is the nxn matrix whose zth column is Aei = ai if i ^ k and Ax = b if i = k. Thus AXk = Mk- Evaluating Xk by cofactor expansion along row k produces det(Xfc) = xk-det(I-i) = xk-Hence by Theorem 4.7, det(Mfc) = det(AXk) = det(A)- det(Xfc) = det(A)-xk-Therefore xfc = [det(A)]-1-det(Mfc). I Example 1 We illustrate Theorem 4.9 by using Cramer's rule to solve the following system of linear equations: xi + 2x2 + 3x3 = 2 x + x3 = 3 x + x2 x3 = 1. The matrix form of this system of linear equations is Ax = b, where '1 l l Because det (A) = 6 ^ 0 , Cramer's rule applies. Using the notation of Theo-rem 4.9, we have Xi = det (Mi) det (A) '2 2 3> 13 0 1 1 1 - 1 det(A) 15 6 x2 = det(M2) det(A) I 1 2 3s 3 1 1 - 1 - 6 det(A) = -1 , 226 and Chap. 4 Determinants det ^3 = det(M3) det(A) det(A) 6 2 Thus the unique solution to the given system of linear equations is (xx,x2,x3) = ( 2 ' - 1 ' 2 In applications involving systems of linear equations, we sometimes need to know that there is a solution in which the unknowns are integers. In this situation, Cramer's rule can be useful because it implies that a system of linear equations with integral coefficients has an integral solution if the determinant of its coefficient matrix is 1. On the other hand, Cramer's rule is not useful for computation because it requires evaluating n + 1 determinants o f n x n matrices to solve a system of n linear equations in n unknowns. The amount of computation to do this is far greater than that required to solve the system by the method of Gaussian elimination, which was discussed in Section 3.4. Thus Cramer's rule is primarily of theoretical and aesthetic interest, rather than of computational value. As in Section 4.1, it is possible to interpret the determinant of a matrix A G M n x n (P ) geometrically. If the rows of A are a i , a 2 , . . . ,a n , respectively, then |det(A)| is the n-dimensional volume (the generalization of area in R2 and volume in R3) of the parallelepiped having the vectors a i , a 2 , . . . ,a as adjacent sides. (For a proof of a more generalized result, see Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, W.H. Freeman and Company, New York, 1993, p. 524.) Example 2 The volume of the parallelepiped having the vectors O] = (1,-2,1), a2 = (1,0, 1), and a3 = (1,1,1) as adjacent sides is = 6. Note that the object in question is a rectangular parallelepiped (see Fig-ure 4.6) with sides of lengths /6. /2, and y/3. Hence by the familiar formula for volume, its volume should be /6*/2*/3 = 6, as the determinant calcu-lation shows. In our earlier discussion of the geometric significance of the determinant formed from the vectors in an ordered basis for R2, we also saw that this Sec. 4.3 Properties of Determinants 227 (1,-2,1) (1,0,-1) Figure 4.6: Parallelepiped determined by three vectors in R3. determinant is positive if and only if the basis induces a right-handed coor-dinate system. A similar statement is true in R'. Specifically, if 7 is any ordered basis for Rn and (3 is the standard ordered basis for Rn, then 7 in-duces a right-handed coordinate system if and only if det(Q) > 0, where Q is the change of coordinate matrix changing 7-coordinates into /^-coordinates. Thus, for instance, 7 = induces a left-handed coordinate system in R3 because whereas 7 = induces a right-handed coordinate system in R3 because r ' 2 det 2 1 0 = 5 > 0. 0 0 1/ 228 Chap. 4 Determinants More generally, if (3 and 7 are two ordered bases for Rn, then the coordinate systems induced by (3 and 7 have the same orientat ion (either both are right-handed or both are left-handed) if and only if det(Q) > 0, where Q is the change of coordinate matrix changing 7-coordinates into /^-coordinates. EXERCISES 1. Label the following statements as true or false. (a) If E is an elementary matrix, then det(F) = 1. (b) For any A,B G Mn X n(F), det(AB) = det(A) det(B). (c) A matrix M G M n x n (F ) is invertible if and only if det(M) = 0. (d) A matrix M G M n X n (F) has rank n if and only if det(M) ^ 0. (e) For any A G M n x n (F ) , det (A*) = - det(A). (f) The determinant of a square matrix can be evaluated by cofactor expansion along any column. (g) Every system of n linear equations in n unknowns can be solved by Cramer's rule. (h) Let Ax = b be the matrix form of a system of n linear equations in n unknowns, where x = (x, x 2 , . . . , x n ) f . If det(A) 7^ 0 and if Mk is the nxn matrix obtained from A by replacing row k of A by bl, then the unique solution of Ax = b is _ det(Mfc) i Q v k _ l 2 Xk~ det(A) tar*-*M.-.n. In Exercises 2-7, use Cramer's rule to solve the given system of linear equa-tions. a-nxi + d2X2 = b 2. a2X + a22x2 = b2 where ana22 - a i 2a 2 i ^ 0 2xi + x2 3x3 = 1 4. x - 2x2 + x3= 0 3xi + 4x2 2x3 = 5 x x2 + Ax3 = 2 6. -8x1 + 3x2 + x3 = 0 2xi - x2 + x3 = 6 2xi + x2 - 3x3 = 5 3. xi - 2x2 + x3 = 10 3x! + 4x2 - 2x3 = 0 xi - x2 + 4x3 = - 4 5. -8x1 + 3x2 + x3 = 8 2xi x2 + x3 = 0 3xi + x2 + x3 = 4 7. - 2 x i - x2 = 1 2 xi + 2x2 + x3 = - 8 8. Use Theorem 4.8 to prove a result analogous to Theorem 4.3 (p. 212), but for columns. 9. Prove that an upper triangular nxn matrix is invertible if and only if all its diagonal entries are nonzero. Sec. 4.3 Properties of Determinants 229 10. A matrix M G M n x n (C) is called nilpotent if, for some positive integer k, Mk = O, where O is the nxn zero matrix. Prove that if M is nilpotent, then det(M) = 0. 11. A matrix M G M n x n (C) is called skew-symmetric if Ml = M. Prove that if M is skew-symmetric and n is odd, then M is not invert-ible. What happens if n is even? 12. A matrix Q G Mn x n(i?) is called orthogonal if QQl I. Prove that if Q is orthogonal, then det(Q) = 1. 13. For M G M n x n(C), let M be the matrix such that (M)ij = ~M~~j for all i,j, where Mij is the complex conjugate of Mij. (a) Prove that det(Af) = det(M). (b) A matrix Q G Mn X n(C) is called unitary if QQ* = / , where Q* = Ql. Prove that if Q is a unitary matrix, then | det(Q)| = 1-14. Let /3 = {ui, u2, , un} be a subset of Fn containing n distinct vectors, and let B be the matrix in M n X n (F) having Uj as column j. Prove that 0 is a basis for Fn if and only if det(B) ^ 0. 15.* Prove that if A, B G M n x n (F ) are similar, then det(A) = det(B). 16. Use determinants to prove that if A, B G M n X n (F) are such that AB =-. / , then A is invertible (and hence B = A - 1 ) . 17. Let A,Be Mnxn(F) be such that AB = -BA. Prove that if n is odd and F is not a field of characteristic two, then A or B is not invertible. 18. Complete the proof of Theorem 4.7 by showing that if A is an elementary matrix of type 2 or type 3, then det(AB) = det(A) det(B). 19. A matrix A G M n x n (F ) is called lower triangular if Atj 0 for 1 < < J < Ti. Suppose that A is a lower triangular matrix. Describe det (A) in terms of the entries of A. 20. Suppose that M G M n X n (F) can be written in the form M = where A is a square matrix. Prove that det(M) = det(A). 21.+ Prove that if M G M n x n ( F ) can be written in the form 'A B' M = O C wheje A and C are square matrices, then det(M) = det(A) det(C). 230 Chap. 4 Determinants 22. Let T: P(F) > F n + 1 be the linear transformation defined in Exer-cise 22 of Section 2.4 by T( / ) = (/(co), / ( d ) , . . . , f(cn)), where^ cn,ci , . . . ,cn are distinct scalars in an infinite field F. Let 0 be the standard ordered basis for P n (F) and 7 be the standard ordered basis for F n + 1 . (a) Show that M = [T]l has the form Co l C Cf] cnJ A matrix with this form is called a Vandermonde matrix. (b) Use Exercise 22 of Section 2.4 to prove that det(M) / 0. (c) Prove that de t (M)= J ] ( < * - * ) . 0Sec. 4.3 Properties of Determinants 231 (b) Show that for 1 < j < n, we have cj2 A = det(A) CjnJ Hint: Apply Cramer's rule to Ax = ej. (c) Deduce that if C is the nxn matrix such that Cij = c,ji, then AC = [det(A)]7. (d) Show that if det(A) / 0, then A'1 = [det(A)]~lC. The following definition is used in Exercises 26 27. Definition. The classical adjoint of a .square matrix A is the transpose of the matrix whose ij-entry is the ij-cofactor of A. 26. Find the classical adjoint of each of the following matrices. '4 0 0> (b) ( 0 4 0 0 0 4, 27. Let C be the classical adjoint of A G M r iXn(F). Prove the following statements. (a) det(C) = [detCA)]'1. (b) Cl is the classical adjoint of A1. (c) If A is an invertible upper triangular matrix, then C and A - 1 are both upper triangular matrices. 28. Let y,y2,- - - ,yn be linearly independent functions in C. For each y G C, define T(y) G C by / y(t) vi(t) y2(t) yn(t) y'(t) . m v'zit) y'Jt) [T(y)](i)=det in) (n). Wn)(t) yn,(t) y}2'(t) y(nt)I 232 Chap. 4 Determinants The preceding determinant is called the Wronskian of y, y,..., yn. (a) Prove that T: C C is a linear transformation. (b) Prove that N(T) contains span({yi, y2, - ,yn})-4.4 SUMMARYIMPORTANT FACTS ABOUT DETERMINANTS In this section, we summarize the important properties of the determinant needed for the remainder of the text. The results contained in this section have been derived in Sections 4.2 and 4.3; consequently, the facts presented here are stated without proofs. The determinant of an n x n matrix A having entries from a field F is a scalar in F , denoted by det(A) or |A|, and can be computed in the following manner: 1. If A is 1 x 1, then det (A) = An, the single entry of A. 2. If A is 2 x 2, then det(A) = AnA2 2 - Ai2A2. For example, d e t ( 5 3) = ( ' 1 ) ( 3 ) ' ( 2 ) ( 5 ) = ' 1 3 -3. If A is n x n for n > 2, then n det(A) = J3 ( -1 )* + 'A W det(Aij) (if the determinant is evaluated by the entries of row i of A) or n det(A) = Y^{-tf+JAij' det( i i j ) i=l (if the determinant is evaluated by the entries of column j of A), where Aij is the (n-l)x(n-l) matrix obtained by deleting row i and column j from A. In the formulas above, the scalar (l)i+J' det(Aij) is called the cofactor of the row i column j entry of A. In this language, the determinant of A is evaluated as the sum of terms obtained by multiplying each entry of some row or column of A by the cofactor of that entry. Thus det (A) is expressed in terms of n determinants of (n 1) x (n 1) matrices. These determinants are then evaluated in terms of determinants of (n 2) x (n 2) matrices, and so forth, until 2 x 2 matrices are obtained. The determinants of the 2 x 2 matrices are then evaluated as in item 2. Sec. 4.4 SummaryImportant Facts about Determinants 233 Let us consider two examples of this technique in evaluating the determi-nant of the 4 x 4 matrix A = (2 1 1 5 1 1 - 4 - 1 2 0 - 3 1 3 6 1 2) To evaluate the determinant of A by expanding along the fourth row. we must know the cofactors of each entry of that row. The cofactor of A.n = 3 is ( - l ) 4 + 1 d e t ( ) , where Let us evaluate this determinant by expanding along the first column. We have det(B) = (-lY^(l)det (l43 ' J ) + ( - l )*+i ( l )dc t ( J , * ) + ( - l ) 3 + 1 ( 0 ) d e t ( ] _ j ) = 1(1)[(-4)(1) - ( - l ) ( - 3 ) ] + (-1)(1)[(1)(1) - (5)(-3)] + 0 = - 7 - 16 + 0 = -23. Thus the cofactor of A4X is ( 1 )5(23) = 23. Similarly, the cofactors of A42, A43, and A44 are 8, 11, and 13. respectively. We can now evaluate the determinant of A by multiplying each entry of the fourth row by its cofactor: t his gives det(A) = 3(23) + 6(8) + 1(11) + 2(-13) = 102. For the sake of comparison, let US also compute the determinant of A by expansion along the second column. The reader should verify that the cofactors of A12, A22, and A42 are 14. 10. and 8, respectively. Thus f - 4 - l (2 1 5N det(A) = ( - l ) 1 + 2 ( l ) d e t 2 - 3 I + ( - l ) 2 + 2 ( l ) d e t 2 - 3 1 3 1 2 / 3 1 2, 2 1 .V + ( - l ) 3 + 2 (0 )de t 1 - 4 - 1 Z 1 2, = 14 + 40 + 0 + 48 = 102. '2 - l ) 4 + ' ( 6 ) d e t I 1 - 4 - 1 2 - 3 1 234 Chap. 4 Determinants Of course, the fact that t he value 102 is obtained again is no surprise since the value of the determinant of A is independent of the choice of row or column used in the expansion. Observe that the computation of det(A) is easier when expanded along the second column than when expanded along the fourth row. The difference is the presence of a zero in the second column, which makes it unnecessary to evaluate one of the cofactors (the cofactor of A32). For this reason, it is beneficial to evaluate the determinant of a mat rix by expanding along a row or column of the matrix that contains t he largest number of zero entries. In fact, it is often helpful to introduce zeros into the matrix by means of elementary row operations before computing the determinant. This technique utilizes the first three properties of the determinant. Properties of the Determinant 1. If B is a matrix obtained by interchanging any two rows or interchanging any two columns of an nxn matrix A, then det(B) = det(A). 2. If B is a matrix obtained by multiplying each entry of some row or column of an n x n matrix A by a scalar k. then det(Z?) = k- det(A). 3. If B is a matrix obtained from an n x n matrix A by adding a multiple of row i to row j or a multiple of column i to column j for i / j. then det(B) = det(A). As an example of the use of these three properties in evaluating deter-minants, let us compute the determinant of the 4 x 4 matrix A considered previously. Our procedure is to introduce zeros into the second column of A by employing property 3, and then to expand along that column. (The elementary row operations used here consist of adding multiples of row 1 to rows 2 and 4.) This procedure yields det(A) = det = 1(-1) (2 1 1 5 1 1 - 4 - 1 2 0 - 3 1 V3 6 1 2 / / - I - 5 l)1 + 2det 2 - 3 V-9 - 5 det ( 2 X - 1 0 2 0 - 9 0 ~6 1 - 2 8 1 - 5 - 3 - 5 5 - 6 1 -28 , The resulting determinant of a 3 x 3 matrix can be evaluated in the same manner: Use type 3 elementary row operations to introduce two zeros into the first column, and then expand along that column. This results in the value 102. Therefore det(A) = l ( - l ) 1 + 2 ( - 1 0 2 ) = 102. Sec. 4.4 SummaryImportant Facts about Determinants 235 Fhe reader should compare this calculation of det (A) with the preceding ones to see how much less work is required when properties 1, 2, and 3 are employed. In the chapters that follow, we often have to evaluate the determinant of matrices having special forms. The next two properties of the determinant are useful in this regard: 4. The determinant of an upper triangular matrix is the product of its diagonal entries. In particular, det(7) = 1. 5. If two rows (or columns) of a matrix are identical, then the determinant of the matrix is zero. As an illustration of property 4. notice that - 3 1 2 0 4 5 = ( - 3 ) ( 4 ) ( - 6 ) = 72. 0 0 - 6 / Property 1 provides an efficient method for evaluating the determinant of a matrix: (a) Use Gaussian elimination and properties 1, 2, and 3 above to reduce the matrix to an upper triangular matrix. (b) Compute the product of the diagonal entries. For instance, del ( I - 1 2 l 2 - 1 - 1 4 - 4 5 - 1 0 - 6 3 - 2 10 -) = det / I - 1 2 0 1 - 5 0 0 3 0 0 9 l - l - 3*6= 18. 1 2 - 4 -*) = det ( 0 0 ^() ( 0 0 V - l I I I - l I 0 0 2 - 5 - 2 4 2 - 5 3 0 1 2 - 2 - 4 / l 2 _4 6 / The next three properties of the determinant are used frequently in later chapters. Indeed, perhaps the most significant property of the determinant is that it provides a simple characterization of invertible matrices. (See prop-erty 7.) 6. For any n x v matrices A and B, det(AB) = det(A)- det(B). 236 Chap. 4 Determinants 7. An n x n matrix A is invertible if and only if det(A) ^ 0. Furthermore, if A is invertible. then det(A - 1) = -r- - . det(A) 8. For any n x n matrix A, the determinants of A and A' are equal. For example, property 7 guarantees that the matrix A on page 233 is invertible because det (A) = 102. The final property, stated as Exercise 15 of Section 4.3, is used in Chap-ter 5. It is a simple consequence of properties 6 and 7. 9. If A and B are similar matrices, then det(A) = det(B). EXERCISES 1. Label the following statements as true or false. (a) The determinant of a square matrix may be computed by expand-ing the matrix along any row or column. (b) In evaluating the determinant of a matrix, it is wise to expand along a row or column containing the largest number of zero en-tries. (c) If two rows or columns of A are identical, then det(A) = 0. (d) If B is a matrix obtained by interchanging two rows or two columns of A, thendet(B) = det(A). (e) If B is a matrix obtained by multiplying each entry of some row or column of A by a scalar, then det(B) = det(A). (f) If B is a matrix obtained from A by adding a multiple of some row to a different row, then det(B) = det(A). (g) The determinant of an upper triangular nxn matrix is the product of its diagonal cut ries. (h) For every A M n x n (F) , det(A') = - det(A). (i) If A, B G MnXfl(F), then det(AB) = det(A) * det(B). (j) If Q is an invertible matrix, then dct(Q~ l) = [det(Q)]-1. (k) A matrix Q is invertible if and only if det(Q) ^ 0. 2. Evaluate the determinant of the following 2 x 2 matrices. (a) (c) 4 - 5 2 3 2 + z - l + 3i 1 - 2i 3 - i (b) (d) - 1 7 3 8 3 ii 0/ li 3. Evaluate the determinant of the following matrices in the manner indi-cated. Sec. 4.4 SummaryImportant Facts about Determinants 237 (a) (c) (e) . : | J 2 3 0/ along the first row 0 1 2s - 1 0 - 3 2 3 0, along the second column / 0 1 + i 2 -2/ 0 1 - i] 3 Ai 0 / along the third row (b) (d) (f) / () (g) 2 1 1 0 - 2 3 - 1 0 V- l 1 2 0 / along the fourth column :S 2 I (h) along the first column 1 0 2N 0 1 o - 1 3 0, along the third row i 2+i 0 - 1 3 2/ 0 - 1 l-i/ along the third column / 1 - 1 2 - 1 - 3 4 1 - 1 2 - 5 - 3 8 - 2 6 - 4 ) along the fourth row 4. Evaluate the determinant of the following matrices by any legitimate method. (h) ( 1 - 2 3 - 1 2 - 5 12 -14 19 - 9 22 -20 31 -A 9 -14 15/ 5. Suppose that M G M n X n (F) can be written in the form A BN M = O / where A is a square matrix. Prove that det(M) = det(A). 238 Chap. 4 Determinants 6.* Prove that if M e M n x n (F ) can be written in the form u (A B M = ( o o)' where A and C are square matrices, then det(Af) = det(A)* det(C). 4.5* A CHARACTERIZATION OF THE DETERMINANT In Sections 4.2 and 4.3, we showed that the determinant possesses a number of properties. In this section, we show that three of these properties completely characterize the determinant; that is, the only function 8: M n x n (F ) * F having these three properties is the determinant. This characterization of the determinant is the one used in Section 4.1 to establish the relationship between det ( ) and the area of the parallelogram determined by u and v. The first of these properties that characterize the determinant is the one described in Theorem 4.3 (p. 212). Definition. A function 8: M n x n ( F ) F is called an n-linear function if it is a linear function of each row of an n x n matrix when the remaining n rows are held fixed, that is. 8 is n-linear if. for every r = 1 ,2 , . . . , n, we have / o.i Op_i u + kv / = 8 ( a, > Or-l li a r+i k8 ( i i a r _i v a r+i / whenever k is a scalar and u, v, and each Oj are vectors in F'. Example 1 The function 8: M x n (F) -> F defined by 8(A) = 0 for each A G M n x n (F ) is an n-linear function. Example 2 For 1 < j F by 8j(A) = AijA2j Anj for each A G M n X n (F) ; that is, 8j(A) equals the product of the entries of column j of Sec. 4.5 A Characterization of the Determinant 239 A. Let Ae M n x n (F) , a{ = (An, Ai2,..., Ain), and v = (bub2,... ,bn) G Fn. Then each 8j is an n-linear function because, for any scalar k, we have a r_i a r + kv a r + i V On / = Aij A ( r_1 ) j(A r j + kbj)A{r+1)j A './ = Aij A(r_,) :,AriA(T.+1) i Anj + Aij A{r_Uj(kbj)A{r+i)j Anj = Aij A(r_i)jArjA(r+i)j Anj + k(Aij A(r_i)j-6jA(r+i)j Anj) ( 1 / ay = 8 ar-i ar a r + i an ) k.8 (lr-l V ar+i an ) Example 3 The function 8: M n X n (F) -> F defined for each A G M n x n (F ) by 8(A) = AnA2 2 -Anil (i.e., 8(A) equals the product of the diagonal entries of A) is an n-linear function. Example 4 The function 8: MnXn(B) > R defined for each A G Mn x n(i?) by 8(A) = tr(A) is not an n-linear function for n > 2. For if / is the nxn identity matrix and A is the matrix obtained by multiplying the first row of / by 2, then 8(A) = n+1^2n = 2-8(1). Theorem 4.3 (p. 212) asserts that the determinant is an n-linear function. For our purposes this is the most important example of an n-linear function. Now we introduce the second of the properties used in the characterization of the determinant. Definition. An n-linear function 8: M n X n (F) > F is called alternating if, for each A G Mn X n(F), we have 8(A) = 0 whenever two adjacent rows of A are identical. 240 Chap. 4 Determinants Theorem 4.10. Let 8: M n X n (F) > F be an alternating n-linear function. (a) If A G M n x n ( F ) and B is a matrix obtained from A by interchanging any two rows of A, then 8(B) = -8(A). (b) If A e M n x n (F ) has two identical rows, then 8(A) = 0. Proof, (a) Let A G M n x n (F ) , and let B be the matrix obtained from A by interchanging rows r and s, where r < s. We first establish the result in the case that s = r + 1. Because 8: M7lXn(F) F is an n-linear function that is alternating, we have 0 = 8 a r + o r + i a r + a r + i = 6' ai a r + a r + i a n / / f l i / ai + Sec. 4.5 A Characterization of the Determinant 241 of A are identical. If .s > r + 1, let B be the matrix obtained from A by interchanging rows r + 1 and s. Then 8(B) = 0 because two adjacent rows of B are identical. But 8(B) = -8(A) by (a). Hence 8(A) = 0. I Corollary 1. Let 8: MnXn(F) F be an alternating n-linear function. If B is a matrix obtained from A G MX(F) by adding a multiple of some row of A to another row, then 8(B) = F be an alternating n-linear function such that 8(1) = 1. For any A, Be M n x n (F ) , we have 8(AB) = 8(A)-8(B). 242 Chap. 4 Determinants Proof. Kxercise. Theorem 4.12. If 8: MXH(F) F is an alternating n-linear function such that 8(1) = 1, then 8(A) = det(A) for every A G M n X n(F) . Proof. Let 8: M n X n (F) > F be an alternating n-linear function such that 8(1) = 1, and let A G M x n (F) . If A has rank less than n, then by Corollary 2 to Theorem 4.10, 8(A) = 0. Since the corollary to Theorem 4.6 (p. 217) gives det(A) = 0, we have 8(A) = det(A) in this case. If, on the other hand, A has rank n, then A is invertible and hence is the product of elementary matrices (Corollary 3 to Theorem 3.6 p. 159), say A = Em---E2En Since 8(1) = 1, it follows from Corollary 3 to Theorem 1.10 and the facts on page 223 that 8(E) det(F) for every elementary matrix E. Hence by Theorems 4.11 and 4.7 (p. 223). we have 8(A) = 8(Em E2EX) = Sec. 4.5 A Characterization of the Determinant 243 3. 8(A) = k, where k is any nonzero scalar 4. 8(A) = A 2 2 5. 8(A) = AiiA23A32 6. 5(A) = A + A 2 3 + A32 7. 5(A) = AnA21A32 8. 8(A) = AnA3iA32 9. 8(A) = AnA222A233 10. 8(A) = AUA22A33 - AUA21A32 11. Prove Corollaries 2 and 3 of Theorem 4.10. 12. Prove Theorem 4.11. 13. Prove that det: M 2 x 2 (F) > F is a 2-linear function of the columns of a matrix. 14. Let a,b,c,d G F. Prove that the function 8: M2X2(F) * F defined by 8(A) = AnA2 2a + AnA2ib + Ai2A22c + Ai2A2id is a 2-linear function. 15. Prove that 8: M2 x 2(F) F is a 2-linear function if and only if it has the form 8(A) = An A22o + AnA2i& + A i 2A22c + Ai2A2xd for some scalars a, b,c,d G F. 16. Prove that if 8: MnXT1(F) > F is an alternating n-linear function, then there exists a scalar k such that 8(A) = A:det(A) for all A G M n X n(F) . 17. Prove that a linear combination of two n-linear functions is an n-linear function, where the sum and scalar product of n-linear functions are as defined in Example 3 of Section 1.2 (p. 9). 18. Prove that the set of all n-linear functions over a field F is a vector space over F under the operations of function addition and scalar mul-tiplication as defined in Example 3 of Section 1.2 (p. 9). 19. Let 8: Mn X n(F) * F be an n-linear function and F a field that does not have characteristic two. Prove that if 8(B) = 8(A) whenever B is obtained from A G M n X n (F) by interchanging any two rows of A, then 8(M) = 0 whenever M G M n X n (F) has two identical rows. 20. Give an example to show that the implication in Exercise 19 need not hold if F has characteristic two. 244 Chap. 4 Determinants INDEX OF DEFINITIONS FOR CHAPTER 4 Alternating n-linear function 239 Angle between two vectors 202 Cofactor 210 Cofactor expansion along the first row 210 Cramer's rule 224 Determinant of a 2 x 2 matrix 200 Determinant of a matrix 210 Left-handed coordinate system 203 n-linear function 238 Orientation of an ordered basis 202 Parallelepiped, volume of 226 Parallelogram determined by two vectors 203 Right-handed coordinate system 202 5 D i a g o n a l i z a t i o n 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalizability 5.3* Matrix Limits and Markov Chains 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem J. his chapter is concerned with the so-called diagonalization problem. For a given linear operator T on a finite-dimensional vector space V, we seek answers to the following questions. 1. Does there exist an ordered basis (3 for V such that [T]/? is a diagonal matrix? 2. If such a basis exists, how can it be found? Since computations involving diagonal matrices are simple, an affirmative answer to question 1 leads us to a clearer understanding of how the operator T acts on V, and an answer to question 2 enables us to obtain easy solutions to many practical problems that can be formulated in a linear algebra context. We consider some of these problems and their solutions in this chapter; see, for example, Section 5.3. A solution to the diagonalization problem leads naturally to the concepts of eigenvalue and eigenvector. Aside from the important role that these concepts play in the diagonalization problem, they also prove to be useful tools in the study of many nondiagonalizable operators, as we will see in Chapter 7. 5.1 EIGENVALUES AND EIGENVECTORS In Example 3 of Section 2.5, we were able to obtain a formula for the reflection of R2 about the line y = 2x. The key to our success was to find a basis (3' for which Jp> is a diagonal matrix. We now introduce the name for an operator or matrix that has such a basis. Definitions. A linear operator T on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis (3 for V such that [T]^ 245 246 Chap. 5 Diagonalization is a diagonal matrix. A square matrix A is called diagonalizable if -A is diagonalizable. We want to determine when a linear operator T on a finite-dimensional vector space V is diagonalizable and, if so, how to obtain an ordered basis (3 = {vi,v2,... ,vn} for V such that [T]^ is a diagonal matrix. Note that, if D = [T]ff is a diagonal matrix, then for each vector Vj G f3, we have J(Vj) = Y D*3Vi - D33V3 = X3V3> i=l where Xj = Djj. Conversely, if 0 = {i>i,i>2,. ,v} is an ordered basis for V such that T(VJ) = XjVj for some scalars Ai, A 2 , . . . , ATl, then clearly [T]3 = /A. 0 0 A2 u 0 0 K) In the preceding paragraph, each vector v in the basis 0 satisfies the condition that T(v) = An for some scalar A. Moreover, because v lies in a basis, v is nonzero. These computations motivate the following definitions. Definitions. Let T be a linear operator on a vector space V. A nonzero vector v G V is called an eigenvector of T if there exists a scalar A such that T(v) = An. The scalar A is called the eigenvalue corresponding to the eigenvector v. Let A be in M n X n (F) . A nonzero vector v G Fn is called an eigenvector of A if v is an eigenvector of LA; that is, if Av = Xv for some scalar X. The scalar X is called the eigenvalue of A corresponding to the eigenvector v. The words characteristic vector and proper vector are also used in place of eigenvector. The corresponding terms for eigenvalue are characteristic value and proper value. Note that a vector is an eigenvector of a matrix A if and only if it is an eigenvector of LA- Likewise, a scalar A is an eigenvalue of A if and only if it is an eigenvalue of L.4. Using the terminology of eigenvectors and eigenvalues. we can summarize the preceding discussion as follows. Theorem 5.1. A linear operator T on a tinilc-dhncnsional vector space V is diagonalizable if and only if there exists an ordered basis 0 for V consisting of eigenvectors ofT. Furthermore, ifT is diagonalizable, 0 = {v,v2,... ,vn} is an ordered basis of eigenvectors of T, and D = [T]^, then D is a diagonal matrix and Djj is the eigenvalue corresponding to Vj for 1 < j < n. Sec. 5.1 Eigenvalues and Eigenvectors 247 To diagonalize a matrix or a linear operator is to find a basis of eigenvec-tors and the corresponding eigenvalues. Before continuing our study of the diagonalization problem, we consider three examples of eigenvalues and eigenvectors. Example 1 Let Since A = U M = 1 3 4 2 1 3 4 2 v = 1 - 1 1 I and v2 -2 2 = - 2 I = -2vi , v is an eigenvector of L4. and hence of A. Here Ai = 2 is the eigenvalue corresponding to v. Furthermore, -A(V2) = 1 3 4 2 = b = 5n2, and so v2 is an eigenvector of LA, and hence of A, with the corresponding eigenvalue A2 = 5. Note that 0 = {vi,v2} is an ordered basis for R2 consisting of eigenvectors of both A and L^, and therefore A and L.,4 are diagonalizable. Moreover, by Theorem 5.1, Me = -2 0 0 5 Example 2 Let T be the linear operator on R2 that rotates each vector in the plane through an angle of 7r/2. It is clear geometrically that for any nonzero vector v, the vectors v and T(v) are not collinear; hence T(v) is not a multiple of v. Therefore T has no eigenvectors and, consequently, no eigenvalues. Thus there exist operators (and matrices) with no eigenvalues or eigenvectors. Of course, such operators and matrices are not diagonalizable. Example 3 Let Ceyo(R) denote the set of all functions f:R>R having derivatives of all orders. (Thus CX(R) includes the polynomial functions, the sine and cosine functions, the exponential functions, etc.) Clearly, C(B) is a subspace of the vector space F(R, R) of all functions from R to R as defined in Section 1.2. Let T: C(B) -> Coc(B) be the function defined by T( / ) = / ' , the derivative of / . It is easily verified that T is a linear operator on C(B). We determine the eigenvalues and eigenvectors of T. 248 Chap. 5 Diagonalization Suppose that / is an eigenvector of T with corresponding eigenvalue A. Then / ' = T( / ) = A/. This is a first-order differential equation whose solu-tions are of the form f(t) = cext for some constant c. Consequently, every real number A is an eigenvalue of T, and A corresponds to eigenvectors of the form cext for c ^ 0. Note that for A = 0, the eigenvectors are the nonzero constant functions. In order to obtain a basis of eigenvectors for a matrix (or a linear opera-tor), we need to be able to determine its eigenvalues and eigenvectors. The following theorem gives us a method for computing eigenvalues. Theorem 5.2. Let A G MnXn(F). Then a scalar X is an eigenvalue of A if and only if det (A - XIn) = 0 . Proof. A scalar A is an eigenvalue of A if and only if there exists a nonzero vector v G Fn such that Av = An, that is, (A - XIn)(v) = 0. By Theorem 2.5 (p. 71), this is true if and only if A XIn is not invertible. However, this result is equivalent to the statement that det(A A/n) = 0. Definition. Let A G Mnxri(F). The polynomial f(t) = det(A - tln) is called the characteristic polynomial l of A. Theorem 5.2 states that the eigenvalues of a matrix are the zeros of its characteristic polynomial. When determining the eigenvalues of a matrix or a linear operator, we normally compute its characteristic polynomial, as in the next example. Example 4 To find the eigenvalues of A = 1 1 4 1 6 M2x2(/2), we compute its characteristic polynomial: det(A - tl2) = det 1 - t 1 - / , = t2 - 2 t - 3 = (t-3)(t + l). It follows from Theorem 5.2 that the only eigenvalues of A are 3 and 1. JThe observant reader may have noticed that the entries of the matrix A ~ tln are not scalars in the field F. They are, however, scalars in another field F(t), the field of quotients of polynomials in t with coefficients from F. Consequently, any results proved about determinants in Chapter 4 remain valid in this context. Sec. 5.1 Eigenvalues and Eigenvectors 249 It is easily shown that similar matrices have the same characteristic poly-nomial (see Exercise 12). This fact enables us to define the characteristic polynomial of a linear operator as follows. Definition. Let T be a linear operator on an n-dimensional vector sjmcc V with ordered basis 0. We define the characteristic polynomial f(t) of T to be the characteristic polynomial of A = [T]^. That is, f(t) = det(A-tln). The remark preceding this definition shows that the definition is indepen-dent of the choice of ordered basis 0. Thus if T is a linear operator on a finite-dimensional vector space V and 0 is an ordered basis for V, then A is an eigenvalue of T if and only if A is an eigenvalue of [T]g. We often denote the characteristic polynomial of an operator T by det(T tl). Example 5 Let T be the linear operator on P2(B) defined by T(f(x)) = f(x)+(x+l)f'(x), let 0 be the standard ordered basis for P2(B), and let A = [T]^. Then The characteristic polynomial of T is (-t 1 0 det(A - tl3) = (let 0 2-t 2 0 0 3 - r = (1 - t)(2 - 0(3 - t) = - ( * - l ) ( * - 2 ) ( t - 3 ) . Hence A is an eigenvalue of T (or A) if and only if A = 1, 2, or 3. Examples 4 and 5 suggest that the characteristic polynomial of an n x n matrix A is a polynomial of degree n. The next theorem tells us even more. It can be proved by a straightforward induction argument. Theorem 5.3. Let A G Mn x n(F). (a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient ( l ) n . (b) A has at most n distinct eigenvalues. Proof. Exercise. 250 Chap. 5 Diagonalization Theorem 5.2 enables us to determine all the eigenvalues of a matrix or a linear operator on a finite-dimensional vector space provided that we can compute the zeros of the characteristic polynomial. Our next result gives us a procedure for determining the eigenvectors corresponding to a given eigenvalue. Theorem 5.4. Let T be a linear operator on a vector space V, and let X be an eigenvalue ofT. A vector v G V is an eigenvector of T corresponding to X if and only if v ^ 0 and v G N(T - AI). Proof. Exercise. Example 6 To find all the eigenvectors of the matrix in Example 4, recall that A has two eigenvalues, Ai = 3 and A2 = 1. We begin by finding all the eigenvectors corresponding to Ai = 3. Let Then is an eigenvector corresponding to Ai = 3 if and only if x ^ 0 and x G N(LR, ); that is, x ^ 0 and - 2 l (xx __ (-2xx + x2 _ /0 4 -2) x2) V 4*i - 2 W V Clearly the set of all solutions to this equation is Hence x is an eigenvector corresponding to Aj = 3 if and only if x = t { for some t ^ 0. Now suppose that x is an eigenvector of A corresponding to A2 = 1. Let , i i (- o^ (2 s B2 = A- X2I = 4 I 0 - 1 / V4 2 Sec. 5.1 Eigenvalues and Eigenvectors Then 'xi 251 1 x2 ' E N ( L f i 2 if and only if rr is a solution to the system 2xi + x2 = 0 4xi + 2x2 = 0. Hence N(Lfl,) = | t ( 2)-teRj. Thus x is an eigenvector corresponding to A2 = 1 if and only if x = t -2 Observe that for some t ^ 0. . - i is a basis for R2 consisting of eigenvectors of A. Thus L^, and hence A, is diagonalizable. Suppose that 0 is a basis for Fn consisting of eigenvectors of A. The corollary to Theorem 2.23 assures us that if Q is the nxn matrix whose columns are the vectors in 0, then Q'1 AQ is a diagonal matrix. In Example 6, for instance, if Q = then - i Q~lAQ = 2 / ' 3 0 0 - 1 Of course, the diagonal entries of this matrix are the eigenvalues of A that correspond to the respective columns of Q. To find the eigenvectors of a linear operator T on an /'.-dimensional vector space, select an ordered basis 0 for V and let A = [T]^. Figure 5.1 is the special case of Figure 2.2 in Section 2.4 in which V = W and 0 = 7. Recall that for v G V, p(v) 252 V V Sec. 5.1 Eigenvalues and Eigenvectors 253 is an eigenvector corresponding to Ai = 1 if and only if x ^ 0 and x G N(LBJ); that is. x is a nonzero solution to the system x2 = 0 x2 + 2x3 = 0 2x3 = 0. Notice that this system has three unknowns, xi, x2, and X3, but one of these, xi, does not actually appear in the system. Since the values of xi do not affect the system, we assign xi a parametric value, say xi = a, and solve the system for x2 and X3. Clearly, x2 = X3 = 0, and so the eigenvectors of A corresponding to Ai = 1 are of the form for a ^ 0. Consequently, the eigenvectors of T corresponding to Ai = 1 are of the form 0^1(aei) = a0^x(ei) = a-1 = a for any a ^ O . Hence the nonzero constant polynomials are the eigenvectors of T corresponding to Ai = 1. Next let A2 = 2, and define ' - 1 1 0N B2 = A - X2I = ( 0 0 2 0 0 1 It is easily verified that and hence the eigenvectors of T corresponding to A2 = 2 are of the form V ^' (ei + e2) = a( l + x) for 0 ^ 0. Finally, consider A3 = 3 and - 2 1 0X A - A 3 / = 0 - 1 2 0 0 0 254 Chap. 5 Diagonalization Since the eigenvectors of T corresponding to A3 = 3 are of the form 0 ^ [ o | 2 | ) = a0 /;1(e i + 2e2 + e3) = a(l + 2x + x2) V W / for a ^ 0. For each eigenvalue, select the corresponding eigenvector with a = 1 in the preceding descriptions to obtain 7 = { l , l + x , l + 2 x + x 2 } , which is an ordered basis for P2(R) consisting of eigenvectors of T. Thus T is diagonalizable. and / l 0 0 [T]7 = 0 2 0 . 0 0 3 / We close this section with a geometric description of how a linear operator T acts on an eigenvector in the context of a vector space V over R. Let v be an eigenvector of T and A be the corresponding eigenvalue. We can think of W = span({n}), the one-dimensional subspace of V spanned by v, as a line in V that passes through 0 and v. For any w G W, w = cv for some scalar c, and hence T(iy) = T(cn) = cT(n) = cAn = Xw; so T acts on the vectors in W by multiplying each such vector by A. There are several possible ways for T to act on the vectors in W, depending on the value of A. We consider several cases. (See Figure 5.2.) CASE 1. If A > 1, then T moves vectors in W farther from 0 by a factor of A. CASE 2. If A = 1, then T acts as the identity operator on W. CASE 3. If 0 < A < 1, then T moves vectors in W closer to 0 by a factor of A. CASE 4. If A = 0, then T acts as the zero transformation on W. CASE 5. If A < 0, then T reverses the orientation of W; that is, T moves vectors in W from one side of 0 to the other. Sec. 5.1 Eigenvalues and Eigenvectors 255 Case 1: A > 1 Case 2: A = 1 Case 3: 0 < A < 1 Case 4: A = 0 Case 5: A < 0 Figure 5.2: The action of T on W = span({v}) when v is an eigenvector of T. To illustrate these ideas, we consider the linear operators in Examples 3, 4, and 2 of Section 2.1. For the operator T on R2 defined by T(ai ,a2) = (ai, a2), the reflection about the x-axis, ei and e2 are eigenvectors of T with corresponding eigen-values 1 and 1, respectively. Since ei and e2 span the x-axis and the y-axis, respectively, T acts as the identity on the x-axis and reverses the orientation of the t/-axis. For the operator T on R2 defined by T(ai ,a2) = (ai,0), the projection on the x-axis, ei and e2 are eigenvectors of T with corresponding eigenvalues 1 and 0, respectively. Thus, T acts as the identity on the x-axis and as the zero operator on the y-axis. Finally, we generalize Example 2 of this section by considering the oper-ator that rotates the plane through the angle 9, which is defined by T0(0.1,0,2) = (ai cos# a2sin*9, ai sin# + a2 cos9). Suppose that 0 < 9 < n. Then for any nonzero vector v, the vectors v and T#(n) are not collinear, and hence T# maps no one-dimensional subspace of R2 into itself. But this implies that T0 has no eigenvectors and therefore no eigenvalues. To confirm this conclusion, let 0 be the standard ordered basis for R2, and note that the characteristic polynomial of T# is det{[Te}3 - tl2) = det cos 9 t sin 1 sin# cos 9 t = t-2- (2cos0)t + l, 256 Chap. 5 Diagonalization which has no real zeros because, for 0 < 9 < it, the discriminant 4 cos2 9 4 is negative. 2. EXERCISES Label the following statements as true or false. (a) Every linear operator on an n-dimensional vector space has n dis-tinct eigenvalues. (b) If a real matrix has one eigenvector, then it has an infinite number of eigenvectors. (c) There exists a square matrix with no eigenvectors. (d) Eigenvalues must be nonzero scalars. (e) Any two eigenvectors are linearly independent. (f) The sum of two eigenvalues of a linear operator T is also an eigen-value of T. (g) Linear operators on infinite-dimensional vector spaces never have eigenvalues. (h) An n x n matrix A with entries from a field F is similar to a diagonal matrix if and only if there is a basis for Fn consisting of eigenvectors of A. (i) Similar matrices always have the same eigenvalues, (j) Similar matrices always have the same eigenvectors, (k) The sum of two eigenvectors of an operator T is always an eigen-vector of T. For each of the following linear operators T on a vector space V and ordered bases 0, compute [T]/?, and determine whether 0 is a basis consisting of eigenvectors of T. (a) V = R2, T 1 0 a - 66 17a - 106 , and 0 = (b) V = Pi (R), J(a + 6x) = (6a - 66) + (12a - 116)x, and 3 = {3 + 4x, 2 + 3x} (c) V = R 3a + 26 - 2c' - 4 a - 36 + 2c . and (d) V = P2(i?),T(a + 6x + cx2) = ( -4a + 26 - 2c) - (7a + 36 + 7c)x + (7a + 6 + 5c>2 , and 0 = {x - x2, - 1 + x2, - 1 - x + x2} Sec. 5.1 Eigenvalues and Eigenvectors (e) V = P3(R), T(a + 6x + ex2 + dx3) = - d + (-c + d)x + (a + 6 - 2c)x2 + ( - 6 + c - 2d)x3, and /? = {1 - x + x 3 , l + x 2 , l , x + x2} 257 (f) V = M2 x 2(i2),T 0 = c d 1 0 1 0 - 7 a - 4 6 + Ac -Ad b - 8 a - 46 + 5c - Ad d , and - 1 2 0 0 1 0 2 0 - 1 0 0 2 3. For each of the following matrices A G M n X n(F) , (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors correspond-ing to A. If possible, find a basis for Fn consisting of eigenvectors of A. If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q~x AQ = D. = R for F = R (hi; (iv; for F = C for F = R 4. For each linear operator T on V, find the eigenvalues of T and an ordered basis 0 for V such that [T]/3 is a diagonal matrix. (a) V = R2 and T(a, 6) = ( -2a + 36. - 10a + 96) (b) V = R3 and T(a, 6, c) = (7a - 46 + 10c, 4a - 36 + 8c, - 2 a + 6 - 2c) (c) V = R3 and T(a, 6, c) = ( -4a + 3 6 - 6c. 6a - 76 + 12c, 6a - 66 + lie) (d) V = P,(7?) and T(u.r + 6) = ( -6a + 26)x + ( -6a + 6) (e) V = P2(R) and T(/(x)) = x/ ' (x) + /(2)x + /(3) (f) V = P3(R) and T(/(x)) = f(x) + /(2)x (g) V = P3(R) and T(/(x)) = xf'(x) + f'(x) - f(2) (h) V = M2x2(R) and T (a ^ 258 7. 8. Chap. 5 Diagonalization c d d 1 a 6 (i) V = M2x2(fi) and T (j) V = M2x2(/?.) and T(A) = A1 + 2 tr(A) I2 5. Prove Theorem 5.4. 6. Let T be a linear operator on a finite-dimensional vector space V, and let 0 be an ordered basis for V. Prove that A is an eigenvalue of T if and only if A is an eigenvalue of [T]#. Let T be a linear operator on a finite-dimensional vector space V. We define the de te rminan t of T, denoted det(T), as follows: Choose any ordered basis 0 for V, and define det(T) = det([T]/3). (a) Prove that the preceding definition is independent of the choice of an ordered basis for V. That is, prove that if 0 and 7 are two ordered bases for V, then det([T]/3) = det([T]7). (b) Prove that T is invertible if and only if det(T) ^ 0. (c) Prove that if T is invertible, then det(T_ 1) = [det(T)]'1. (d) Prove that if U is also a linear operator on V, then det(TU) = dct(T)-det(U). (e) Prove that det(T Aly) = det([T]/j XI) for any scalar A and any ordered basis 0 for V. (a) Prove that a linear operator T on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of T. (b) Let T be an invertible linear operator. Prove that a scalar A is an eigenvalue of T if and only if A~! is an eigenvalue of T - 1 . (c) State and prove results analogous to (a) and (b) for matrices. 9. Prove that the eigenvalues of an upper triangular matrix M are the diagonal entries of M. 10. Let V be a finite-dimensional vector space, and let A be any scalar. (a) For any ordered basis 0 for V, prove that [Aly]/? = XI. (b) Compute the characteristic polynomial of Aly. (c) Show that Aly is diagonalizable and has only one eigenvalue. 11. A scalar mat r ix is a square matrix of the form XI for some scalar A; that is, a scalar matrix is a diagonal matrix in which all the diagonal entries are equal. (a) Prove that if a square matrix A is similar to a scalar matrix XI, then A- XL (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix. Sec. 5.1 Eigenvalues and Eigenvectors 259 (c) Prove that I . 1 is not diagonalizable. 12. (a) Prove that similar matrices have the same characteristic polyno-mial. (b) Show that the definition of the characteristic polynomial of a linear operator on a finite-dimensional vector space V is independent of the choice of basis for V. 13. Let T be a linear operator on a finite-dimensional vector space V over a field F, let 0 be an ordered basis for V, and let A = [T]/3. In reference to Figure 5.1, prove the following. (a) If r G V and 0(v) is an eigenvector of A corresponding to the eigenvalue A, then v is an eigenvector of T corresponding to A. (b) If A is an eigenvalue of A (and hence of T), then a vector y G Fn is an eigenvector of A corresponding to A if and only if 4>~Q (y) is an eigenvector of T corresponding to A. 14.* For any square matrix A, prove that A and A1 have the same charac-teristic polynomial (and hence the same eigenvalues). 15- (a) Let T be a linear operator on a vector space V, and let x be an eigenvector of T corresponding to the eigenvalue A. For any posi-tive integer m, prove that x is an eigenvector of T m corresponding to the eigenvalue A7', (b) State and prove the analogous result for matrices. 16. (a) Prove that similar matrices have the same trace. Hint: Use Exer-cise 13 of Section 2.3. (b) How would you define the trace of a linear operator on a finite-dimensional vector space? Justify that your definition is well-defined. 17. Let T be the linear operator on MnXn(i?) defined by T(A) = A1. (a) Show that 1 are the only eigenvalues of T. (b) Describe the eigenvectors corresponding to each eigenvalue of T. (c) Find an ordered basis 0 for M2x2(Z?) such that [T]/3 is a diagonal matrix. (d) Find an ordered basis 0 for Mnx(R) such that [T]p is a diagonal matrix for n > 2. 18. Let A,B e Mx n(C). (a) Prove that if B is invertible, then there exists a scalar c G C such that A + cB is not invertible. Hint: Examine det(A + cB). r 260 Chap. 5 Diagonalization (b) Find nonzero 2 x 2 matrices A and B such that both A and A + cB are invertible for all e G C. 19. * Let A and B be similar nxn matrices. Prove that there exists an n-dimensional vector space V, a linear operator T on V, and ordered bases 0 and 7 for V such that A = [T]/3 and B = [T]7. Hint: Use Exercise 14 of Section 2.5. 20. Let A be an nxn matrix with characteristic polynomial f(t) = (-l)ntn + an - i r ' - 1 + + a ^ + a0. Prove that /(0) = an = det(A). Deduce that A is invertible if and only if a0 ^ 0. 21. Let A and f(t) be as in Exercise 20. (a) Prove t h a t / ( 0 = (An-t)(A22-t) (Ann-t) + q(t), where q(t) is a polynomial of degree at most n 2. Hint: Apply mathematical induction to n. (b) Show that tr(A) = ( - l ) n - 1 a n _ i . 22.t (a) Let T be a linear operator on a vector space V over the field F, and let g(t) be a polynomial with coefficients from F. Prove that if x is an eigenvector of T with corresponding eigenvalue A, then g(T)(x) = g(X)x. That is, x is an eigenvector of o(T) with corre-sponding eigenvalue g(X). (b) State and prove a comparable result for matrices. (c) Verify (b) for the matrix A in Exercise 3(a) with polynomial g(t) = 2t2 t + 1, eigenvector x = I J, and corresponding eigenvalue A = 4. 23. Use Exercise 22 to prove that if f(t) is the characteristic polynomial of a diagonalizable linear operator T, then / (T) = To, the zero opera-tor. (In Section 5.4 we prove that this result does not depend on the diagonalizability of T.) 24. Use Exercise 21(a) to prove Theorem 5.3. 25. Prove Corollaries 1 and 2 of Theorem 5.3. 26. Determine the number of distinct characteristic polynomials of matrices in M2x2(Z2). Sec. 5.2 Diagonalizability 261 5.2 DIAGONALIZABILITY In Section 5.1, we presented the diagonalization problem and observed that not all linear operators or matrices are diagonalizable. Although we are able to diagonalize operators and matrices and even obtain a necessary and suf-ficient condition for diagonalizability (Theorem 5.1 p. 246), we have not yet solved the diagonalization problem. What is still needed is a simple test to determine whether an operator or a matrix can be diagonalized, as well as a method for actually finding a basis of eigenvectors. In this section, we develop such a test and method. In Example 6 of Section 5.1, we obtained a basis of eigenvectors by choos-ing one eigenvector corresponding to each eigenvalue. In general, such a procedure does not yield a basis, but the following theorem shows that any set constructed in this manner is linearly independent. Theorem 5.5. Let T be a linear operator on a vector space V, and let Ai, A2 , . . . , Xk be distinct eigenvalues of T. Ifvi,v2, ,vk are eigenvectors of T such that Xi corresponds to Vi (I < i < k), then {vi,v2,... ,vk} is linearly independent. Proof. The proof is by mathematical induction on k. Suppose that k = 1. Then vi ^ 0 since v is an eigenvector, and hence {vi} is linearly independent. Now assume that the theorem holds for k - 1 distinct eigenvalues, where k 1 > 1, and that we have k eigenvectors v, v2,..., Vk corresponding to the distinct eigenvalues Ai, A 2 , . . . , Xk. We wish to show that {vi, v2,..., vk} is linearly independent. Suppose that a i , a 2 , . . . , a k are scalars such that a i n i+a 2 n 2 - l Yakvk = 0. (1) Applying T - A^ I to both sides of (1), we obtain ai(Ai - Xk)vi +a2(A2 - Afc)n2 + +afc_i(Afe_1 - Xk)vk-i = 0. By the induction hypothesis {f i ,v 2 , . . . ,vk-i} is linearly independent, and hence ai(Xi - Afc) = a2(A2 - Afc) = = afc_i(Afc-i - Afc) = 0. Since Ai, A2 , . . . , A^ are distinct, it follows that Aj - A^ ^ 0 for 1 < i < k 1. So ai = a2 = = afc_i = 0, and (1) therefore reduces to akvk = 0. But vk ^ 0 and therefore ak = 0. Consequently ai = a2 = = ak = 0, and it follows that {vi,v2,... ,vk} is linearly independent. Corollary. Let T be a linear operator on an n-dimensional vector space V. If T has n distinct eigenvalues, then T is diagonalizable. 262 Chap. 5 Diagonalization Proof. Suppose that T has n distinct eigenvalues Ai, . . . ,A. For each i choose an eigenvector Vi corresponding to A;. By Theorem 5.5, {v,... ,vn} is linearly independent, and since dim(V) = n, this set is a basis for V. Thus, by Theorem 5.1 (p. 246), T is diagonalizable. 1 Example 1 Let A = 1 1 1 1 G M2x2(R). The characteristic polynomial of A (and hence of L^) is det (A - tl) = det - t 1 1 1-rf = t(t - 2), and thus the eigenvalues of L^ are 0 and 2. Since L^ is a linear operator on the two-dimensional vector space R2, we conclude from the preceding corollary that L.4 (and hence A) is diagonalizable. The converse of Theorem 5.5 is false. That is, it is not true that if T is diagonalizable. then it has n distinct eigenvalues. For example, the identity operator is diagonalizable even though it has only one eigenvalue, namely, A = l. We have seen that diagonalizability requires the existence of eigenvalues. Actually, diagonalizability imposes a stronger condition on the characteristic polynomial. Definition. A polynomial f(t) in P(F) splits over F if there are scalars c , a i , . . . ,a (not necessarily distinct) in F such that f(t) = c(t-ai)(t-a2)---(t-an). For example, t2 - 1 = (t + l)(t - 1) splits over R, but (t2 + l)(t - 2) does not split over R because t2 + 1 cannot be factored into a product of linear factors. However, (t2 + )(t - 2) does split over C because it factors into the product (t + i)(t i)(t 2). If f(t) is the characteristic polynomial of a linear operator or a matrix over a field F , then the statement that f(t) splits is understood to mean that it splits over F. Theorem 5.6. The characteristic polynomial of any diagonalizable linear operator splits. Proof. Let T be a diagonalizable linear operator on the n-dimensional vector space V. and let 0 be an ordered basis for V such that [T)p = D is a Sec. 5.2 Diagonalizability diagonal matrix. Suppose that D = 263 (Xx 0 0 A2 0 0 o o A , , / and let f(t) be the characteristic polynomial of T. Then f(t) = det(D - tl) = det (Xx-t 0 0 0 A2 - f 0 V 0 0 A - t / = (Ai - t)(X2 - t) (A - 1 ) = ( - 1 H * - Ai)(t - A2) . . . (t - An). I From this theorem, it is clear that if T is a diagonalizable linear operator on an n-dimensional vector space that fails to have distinct eigenvalues, then the characteristic polynomial of T must have repeated zeros. The converse of Theorem 5.6 is false: that is, the character ist i< polynomial of T may split, but T need not be diagonalizable. (See Example 3. which follows.) The following concept helps us determine when an operator whose characteristic polynomial splits is diagonalizable. Definition. Let X be an eigenvalue of a linear operator or matrix with characteristic polynomial f(t). The (algebraic) multiplicity of X is the largest positive integer k for which (t - X)k is a factor of f(t). Example 2 Let which has characteristic polynomial f(t) = (t 3)2(t A). Hence A = 3 is an eigenvalue of A with multiplicity 2. and A = 4 is an eigenvalue of A with multiplicity 1. If T is a diagonalizable linear operator on a finite-dimensional vector spate V, then there is an ordered basis 0 for V consisting of eigenvectors of T. We know from Theorem 5.1 (p. 246) that [T]/3 is a diagonal matrix in which the diagonal entries are the eigenvalues of T. Since the characteristic polynomial of T is detQT]/? tl), it is easily seen that each eigenvalue of T must occur as a diagonal entry of [T]/3 exactly as many times as its multiplicity. Hence 264 Chap. 5 Diagonalization 0 contains as many (linearly independent) eigenvectors corresponding to an eigenvalue as the multiplicity of that eigenvalue. So the number of linearly independent eigenvectors corresponding to a given eigenvalue is of interest in determining whether an operator can be diagonalized. Recalling from Theo-rem 5.4 (p. 250) that the eigenvectors of T corresponding to the eigenvalue A are the nonzero vectors in the null space of T AI, we are led naturally to the study of this set. Definition. Let T be a linear operator on a vector space V, and let X be an eigenvalue ofT. Define E = {x G V: T(x) = Ax} = N(T Aly). The set E is called the eigenspace of T corresponding to the eigenvalue X. Analogously, we define the eigenspace of a square matrix A to be the eigenspace of LA. Clearly, E is a subspace of V consisting of the zero vector and the eigen-vectors of T corresponding to the eigenvalue A. The maximum number of linearly independent eigenvectors of T corresponding to the eigenvalue A is therefore the dimension of EA- Our next result relates this dimension to the multiplicity of A. Theorem 5.7. Let T be a linear operator on a finite-dimensional vec-tor space V, and let X be an eigenvalue of T having multiplicity m. Then 1 < dim(EA) < m. Proof. Choose an ordered basis {v. r2 vp} for EA. extend it to an or-dered basis 0 = {vi, t)2, , Up, vp+i,..., vn} for V, and let A = [T]g. Observe that V{ (1 < i < p) is an eigenvector of T corresponding to A, and therefore A-{() C By Exercise 21 of Section 4.3, the characteristic polynomial of T is B C - tln-p f{t) = det(A - tin) = det ({X J)l' = det((A - t)Ip) det(C - tln-p) = (X-ty>g(t), where g(t) is a polynomial. Thus (A t)p is a factor of f(t), and hence the multiplicity of A is at least p. But dim(EA) = p, and so dim(EA) < rn. Example 3 Let T be the linear operator on P2(R) defined by T(/(x)) = f'(x). The matrix representation of T with respect to the standard ordered basis 0 for Sec. 5.2 Diagonalizability P2(R.) is /.) 1 0 [J}0 = 0 0 2 0 0 0 / Consequently, the characteristic polynomial of T is / - / . 1 0N det([T]p - /) = det 0 - / 2 V 0 0 -t, 265 = -t*. Thus T has only one eigenvalue (A = 0) wit h mult iplicity 3. Solving T(/(.r)) = f'(x) = 0 shows that EA = N(T - AI) = N(T) is the subspace of P2(R) con-sisting of the constant polynomials. So {1} is a basis for EA, and therefore dim(EA) = 1. Consequently, there is no basis for P2(R) consisting of eigen-vectors of T, and therefore T is not diagonalizable. Example 4 Let T be the linear operator on R3 defined by A; A /4a i + a 3 T a2 I = I 2ai + 3a2 + 2o3 . ' . i / ai + Au->) We determine the eigenspace of T corresponding to each eigenvalue. Let > be the standard ordered basis for R3. Then /4 0 l [T],,= I 2 3 2 . and hence the characteristic polynomial of T is / 4 - 1 0 1 N det([T]/3 - tl) = (let [ 2 3 - t 2 I 0 A-t - ( t - 5 ) ( t - 3 ) 2 . So the eigenvalues of T are Ai = 5 and A2 = 3 with multiplicities 1 and 2, respectively. 266 Chap. 5 Diagonalization EAX is the solution space of the system of linear equations - x i + x3 = 0 2xi - 2x2 + 2x3 = 0 xi - x3 = 0. It is easily seen (using the techniques of Chapter 3) that is a basis for EA, Hence dim(EA,) = 1. Similarly, EA2 = N(T A2I) is the solution space of the system x + x3 = 0 2xi + 2x3 = 0 x + x3 = 0. Since the unknown x2 does not appear in this system, we assign it a para-metric value, say, X2 = s, and solve the system for xi and x3, introducing another parameter t. The result is the general solution to the system 0 , for s, t G R. It follows that is a basis for EA2, and dim(EA2) = 2. In this case, the multiplicity of each eigenvalue Ai is equal to the dimension of the corresponding eigenspace EA^ . Observe that the union of the two bases just derived, namely, is linearly independent and hence is a basis for R3 consisting of eigenvectors of T. Consequently, T is diagonalizable. Sec. 5.2 Diagonalizability 267 Examples 3 and 1 suggest that an operator whose characteristic polyno-mial splits is diagonalizable if and only if the dimension of each eigenspace is equal to the multiplicity of the corresponding eigenvalue. This is indeed true, as we now show. We begin with the following lemma, which is a slight variation of Theorem 5.5. Lemma. Let T be a linear operator, and let Ai,A2 , AA: be distinct eigenvalues of T. For each i = 1,2,...,A;, let Vi G EAJ, the eigenspace corre-sponding to A,. //' Vi+v2- rVk = 0, then V{ = 0 for all i. Proof. Suppose otherwise. By renumbering if necessary, suppose that, for 1 < m < k, we have v, ^ 0 for 1 < i < m, and Vi = 0 for / > m. Then, for each i < m, Vi is an eigenvector of T corresponding to A, and v + v2 H h vm = 0. But this contradicts Theorem 5.5, which states that these r,'s are linearly independent. We conclude, therefore, that Vi = 0 for all i. Theorem 5.8. Let T be a linear operator on a vector space V. and let X, A2 , . . . . A/,, be distinct eigenvalues ot'T. For each i = 1,2,... ,k. let S, be a finite linearly independent subset of the eigenspace EA, Then S = Si U S2 u U Sk is a linearly independent subset ofW. Proof. Suppose that for each i Si = {viUVi2,...,vini}. Then S = [vy : 1 < j < /;,. and 1 < i < A:}. Consider an- scalars {a^} such that ^^a-ijVij = 0. i=i j=i For each i, let Wi = ^ajjVij. Then Wi G EA, for each /'. and W + + wk = 0. Therefore, by the lemma. Wi 0 for all i. Bui each .S', is linearly independent, and hence a^ 0 for all j . We conclude that S is lineally independent. 268 Chap. 5 Diagonalization Theorem 5.8 tells us how to construct a linearly independent subset of eigenvectors, namely, by collecting bases for the individual eigeuspaces. The next theorem tells us when the resulting set is a basis for the entire space. Theorem 5.9. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits. Let Ai, A 2 , . . . . Xk be the distinct eigenvalues ofT. Then (a) T is diagonalizable if and only if the multiplicity of Xi is equal to dim(EA,) for all i. (b) If T is diagonaliy.nhie and I, is an ordered basis for EA, for each i, then 0 = 0i U02 U U0k is an ordered basis2 for V consisting of eigenvectors ofT. Proof. For each /. let /;;, denote the multiplicity of A,, d, = dim(EA,), and n = dim(V). First, suppose that T is diagonalizable. Let 0 be a basis for V consisting of eigenvectors of T. For each /. let 0r = 0 n Et, the set of vectors in 0 that are eigenvectors corresponding to At, and let n^ denote the number of vectors in ).,. Then n, < 0 for all i, we conclude that ntj = dt- for all i. Concrsely. suj)pose that nij = d,; for all i. We simultaneously show that T is diagonalizable and prove (b). For each i, let 0i be an ordered basis for EA< , and let 0 = 0 J02U U0k. By Theorem 5.8, 0 is linearly independent. Furthermore, since d, = m, for all i. 0 contains i=l i=l = n 2 We regard 0 U 02 U l)0k as an ordered basis in the natural way- -the vectors in 0 are listed first (in the same order as in f3), then the vectors in /52 (in the same order as in ii->), etc. Sec. 5.2 Diagonalizability 269 vectors. Therefore 3 is an ordered basis for V consisting of eigenvectors of V. and we conclude that T is diagonalizable. This theorem completes our study of the diagonalization problem. We summarize our results. Test for Diagonalization Let T be a linear operator on an n-dimensional vector space V. Then T is diagonalizable if and only if both of the following conditions hold. 1. The characteristic polynomial ofT splits. 2. For each eigenvalue A ofT. the multiplicity of A equals n - rank(T AI). These same conditions can be used to test if a square matrix A is diagonal-izable because diagonalizability of A is equivalent to diagonalizability of the operator L^. If T is a diagonalizable operator and :j. 02..... 0k are ordered bases for the eigenspaces of T. then the union 0 = 0iU02U- -U0k is an ordered basis for V consisting of eigenvectors of T. and hence [T]@ is a diagonal matrix. When testing T for diagonalizability, it is usually easiest to choose a conve-nient basis Q for V and work with B = [T]a. If the characteristic polynomial of B splits, then use condition 2 above to check if the multiplicity of each repeated eigenvalue of B equals n - rak(B Xf). (By Theorem 5.7, condi-tion 2 is automatically satisfied for eigenvalues of multiplicity 1.) If so, then B. and hence T, is diagonalizable. If T is diagonalizable and a basis 8 for V consisting of eigenvectors o fT is desired, then we first find a basis for each eigenspace of B. The union of these bases is a basis *) for F' consisting of eigenvectors of B. Each vector in 7 is the coordinate vector relative to a of an eigenvector of T. The set consisting of these n eigenvectors of T is the desired basis >. Furthermore, if A is an n x n diagonalizable matrix, we can use the corol-lary to Theorem 2.23 (p. 115) to find an invertible n x n matrix Q and a diagonal n x n matrix D such that Q~l AQ = D. The matrix Q has as its columns the vectors in a basis of eigenvectors of A. and I) has as its j t h diagonal entry the eigenvalue of A corresponding to the jth column of Q. We now consider some examples illustrating the preceding ideas. Example 5 We test t he matrix G M3x:i(/r) for diagonalizability. 270 Chap. 5 Diagonalization The characteristic polynomial of A is de t (A- t J ) = -(t A)(t-3)2, which splits, and so condition 1 of the test for diagonalization is satisfied. Also A has eigenvalues Ai = 4 and A2 = 3 with multiplicities 1 and 2. respectively. Since A| has multiplicity 1, condition 2 is satisfied for X. Thus we need only test condition 2 for A2. Because /() 1 0 A - X2I = I 0 0 0 I 7 has rank 2, we see that 3 rank(A A27) = 1. which is not the multiplicity of A2. Thus condition 2 fails for A2, and A is therefore not diagonalizable. Example 6 Let T be the linear operator on P2(R) defined by T(/(x)) = / ( l ) + f'(0)x + (/'(()) + /'(0))x2 . We first test T for diagonalizability. Let a denote the standard ordered basis for P2(R) and B = [T)n. Then / l 1 l B = 0 1 0 . 0 1 l) The characterist ic polynomial of li, and hence of T, is (/ I )2{l 2), which splits. Hence condition 1 of the test for diagonalization is satisfied. Also B has the eigenvalues Ai = 1 and A2 = 2 with multiplicities 2 and 1, respectively. Condition 2 is satisfied for A2 because it has multiplicity 1. So we need only verify condition 2 for Ai = 1 . For this case, 3 - rank( - A, I) = 3 - rank which is equal to the multiplicity of A|. Therefore T is diagonalizable. We now find an ordered basis 7 for R3 of eigenvectors of B. We consider each eigenvalue separately. The eigenspace corresponding to Ai = 1 is Ex, = G R' 0 0 0 1 0 1 1 0 1/ (xi u V':i = 0 Sec. 5.2 Diagonalizability which is the solution space for the system x2 + 2:3 = 0, and has 271 7i = tr as a basis. The eigenspace corresponding to A2 = 2 is EA2 = X2 GR3 : - 1 1 V 0 - 1 0 0 1 (I, which is the solution space for the system - x i + x2 + x3 = 0 X2 = 0 , and has as a basis. Let 72 = 7 = 7i U 72 = ' tr i r Then 7 is an ordered basis for R-i consisting of eigenvectors of B. Finally, observe that the vectors in 7 are the coordinate vectors relative to a of the vectors in the set 0={l,-x + x2,l+x2}, which is an ordered basis for P2(R) consisting of eigenvectors of T. Thus / l 0 0 [T}0 = 0 1 0 . o 0 2 / 272 Chap. 5 Diagonalization Our next example is an application of diagonalization that is of interest in Section 5.3. Example 7 Let A = 0 - 2 1 3 We show that A is diagonalizable and find a 2 x 2 matrix Q such that Q~1AQ is a diagonal matix. We then show how to use this result to compute An for any positive integer n. First observe that the characteristic polynomial of A is (t l)(t 2), and hence A has two distinct eigenvalues, Ai = 1 and A2 = 2. By applying the corollary to Theorem 5.5 to the operator L^. we see that A is diagonalizable. Moreover, 7 l = { C ~ i ) } a n d * ' ' { ( ~ i , are bases for the eigenspaces EAX and EA2, respectively. Therefore 7 = 7i U 72 - 2 1 is an ordered basis for R2 consisting of eigenvectors of A. Let - 2 - T Q = 1 1 the matrix whose columns are the vectors in 7. Then, by the corollary to Theorem 2.23 (p. 115), D = Q'lAQ = [LA}0 = 1 0 0 2) ' To find An for any positive integer n, observe that A = QDQ 1. Therefore An = (QDQ'1)n = (QDQ-l)(QDQ-')- (QDQ-1) = QDnQ~l l o 2I -2 -l)(l i ) H ~l 1 l) l() 2n 1 2 2 _ 2n 2- 2 n + 1 - l + 2 n - l + 2 n + 1 Sec. 5.2 Diagonalizability 273 We now consider an application that uses diagonalizat ion to solve a system of differential equations. Systems of Differential Equations Consider the system of differential equations x = 3Xi + X2 + X3 Xo = 2xi + 4x2 + 2x3 x3 = - x i - x2 + x3, where, for each i, Xj = Xi(t) is a differentiable real-valued function of the real variable t. Clearly, this system has a solution, namely, the solution in which each Xi(t) is the zero function. We determine all of the solutions to this system. Let x: R > R3 be the function defined by The derivative of x, denoted x', is defined by /'. (t) At) = A(t) V'3(t)) Let A = be the coefficient matrix of the given system, so that we can rewrite the system as the matrix equation x' = Ax. It can be verified that for Q = 0 1 0 - 1 1 - M -2 1 and D = (2 0 o 0 2 0 0 0 1 we have Q~lAQ = D. Substitute A = QDQ~l into x' = Ax to obtain x' = QDQ~lx or, equivalently, Q~xx' = DQ~lx. The function y: R -> R3 defined by y(t) = Q~lx(t) can be shown to be differentiable, and y' = Q~lx' (see Exercise 16). Hence the original system can be written as y' = Dy. 274 Chap. 5 Diagonalization Since D is a diagonal matrix, the system y' Dy is easy to solve. Setting y(t) = fyi{t) V2{t) we can rewrite y' = Dy as (v(i) The three equations f2yi(t)' 2lte(0 y[ = 2j/i y2 = 22/2 y3 = 4i/3 are independent of each other, and thus can be solved individually. It is easily seen (as in Example 3 of Section 5.1) that the general solution to these equations is yi(t) = cie2 ' , y2(t) = c2e2t, C3 are arbitrary constants. Finally, and y3(t) c3e4t, where ci,c2, and ( * i ( 0 M O x3(t)J = x(t) = Qy(t) = / - I ' 0 - 1 1 - 2 1 cie c2e ,c3e 2t 2t At cie 2t c3e i C]t - c2e2t - 2e3e4t 21 + c2c2' + c3e4t yields the general solution of the original system. Note that this solution can be written as x(t) = ,2t + f ,4t The expressions in brackets are arbitrary vectors in EA, and EA2, respectively, where Ai = 2 and A2 = 4. Thus the general solution of the original system is x(t) = e2tzi + e4tz2, where z G EA, and z2 G EA2. This result is generalized in Exercise 15. Direct Sums* Let T be a linear operator on a finite-dimensional vector space V. There is a way of decomposing V into simpler subspaces that offers insight into the Sec. 5.2 Diagonalizability 275 behavior of T. This approach is especially useful in Chapter 7, where we study nondiagonalizable linear operators. In the case of diagonalizable operators, the simpler subspaces are the eigenspaces ol the operator. Definition. Let Wi, W 2 . . . . , Nk be .subspaces of a vector space V. We define the sum of these subspaces to be the set {vi + v2 + r vk: Vi G Wi for 1 < i < k), k which we denote by W] + W2 - + WA or W,. It is a simple exercise to show that the sum of subspaces of a vector space is also a subspace. Example 8 Let V = R3, let W] denote the xy-pane, and let W2 denote the yz-plane. Then R3 = Wi + W2 because, for any vector (a, b, c) G R3. we have (a,6,c) = (a,0,0) + (0.fr,c), where (a, 0,0) G W, and (0, b, c) G W2. Notice that in Example 8 the representation of (a, b, c) as a sum of vectors in Wi and W2 is not unique. For example, (a, b, c) = (a, b, 0) + (0,0. c) is another representation. Because we are often interested in sums for which representations are unique, we introduce a condition that assures this out-come. The definition of direct sum that follows is a generalization of the definition given in the exercises of Section 1.3. Definition. Let V/i,N2 W/, be subspaces of a vector space V. We call V the direct sum of the subspaces Wi.W2 vVk and write V = W, @ W 2 - - - 0 V V A . . , if and v = f > i=i W, n Y^ W, = {0} for each j (1 < j < k). Example 9 Let V = R', W, = {(a,6,0,0): a,b, R}, W2 = {(0,0,c,0): c G R}, and W3 = {(0,0,0, d): d G R}. For any (a, b, c, d) G V. (a, 6, c, d) = (a, 6,0,0) + (0,0, c, 0) + (0,0,0, d) Wi + W2 + W3. 276 Thus Chap. 5 Diagonalization V = ^ W To show that V is the direct sum of Wi, W2, and W3, we must prove that Wi n (W2 + W3) = W2 n (Wi + W3) = W3 n (Wi + W2) = {0}. But these equalities are obvious, and so V = Wi W2 W3. Our next result contains several conditions that are equivalent to the definition of a direct sum. Theorem 5.10. Let Wi, W 2 , . . . . Wfc be subspaces of a finite-dimensional vector space V. The following conditions are equivalent. (a) v = w 1 e w 2 - - - Wfc. k (b) V BB V^Wi and, for any vectors vi,v2,--- ,vk such that Vi G Wj J=I (1 * > > , . i=l Now suppose that i>i,n2,... ,vk are vectors such that Vi G W^ for all i and V + V2 H ~vk = 0. Then for any j ~v3 B l e 5 Z W i ' But Vj G Wj and hence - ^ e W j - n ^ w ^ i o } . So Vj = 0, proving (b). Now assume (b). We prove (c). Let v G V. By (b), there exist vectors vi,i>2, . . , vk such that V{ G W, and v = vi + v2 + + vk. We must show Sec. 5.2 Diagonalizability 277 that this representation is unique. Suppose also that v = W +W2 H h wk, where Wi G Wj for all i. Then (vi - w^ + (v2-w2) + --- + (vk - wk) = 0. But Vi Wi G Wi for all i, and therefore Vi Wi 0 for all i by (b). Thus V< = Wi for all 2, proving the uniqueness of the representation. Now assume (c). We prove (d). For each i, let 7, be an ordered basis for Wt. Since k v = ] T w , i=i by (c), it follows that 71 U 72 U U 7fc generates V. To show that this set is linearly independent, consider vectors Vij G 7< (j = 1,2, . . . ,m* and i = 1 ,2 , . . . , k) and scalars a^ such that ^OijVij = 0. 1,3 For each i, set Wi = JJ ^ j Then for each i, Wi G span(7i) = Wi and n?i + w2 H 1- tnfc = 2 J ai j% = 0. .j Since 0 G Wi for each i and ^ + 0 H h ^ = wi + if2 H -u)k, (c) implies that Wi 0 for all i. Thus 0 = W = 7. aijvij j = for each i. But each 7^ is linearly independent, and hence a^ = 0 for all i and j. Consequently 71 U 72 U U 7fc is linearly independent and therefore is a basis for V. Clearly (e) follows immediately from (d). Finally, we assume (e) and prove (a). For each i, let 7< be an ordered basis for Wi such that 71 U 72 U U 7fc is an ordered basis for V. Then V = span(7i U72U---U7fc) 278 Chap. 5 Diagonalization k = span(7i) + span(72) + + span(7fc) = J j W,-i=i by repeated applications of Exercise 14 of Section 1.4. Fix j (1 < j < k), and suppose that, for some nonzero vector v V, Then j e W j - n ^ w , - . G Wj = span(7j) and w G 2_,Wi = span I ^J i * j Lj Hence n is a nontrivial linear combination of both 7j and so that v can be expressed as a linear combination of 71 U 72 U U 7fc in more than one way. But these representations contradict Theorem 1.8 (p. 43), and so we conclude that w , n ] T w ( = {0}. proving (a). I With the aid of Theorem 5.10, we are able to characterize diagonalizability in terms of direct sums. Theorem 5.11. A linear operator T on a finite-dimensional vector space V is diagonalizable if and only if/ is the direct sum of the cigenspaces ofT. Proof. Let Ai, A 2 , . . . . Afc be the distinct eigenvalues of T. First suppose that T is diagonalizable, and for each i choose an ordered basis 7i for the eigenspace EA^ By Theorem 5.9, 71 U 72 U U 7fc is a basis for V, and hence V is a direct sum of the EA '^S by Theorem 5.10. Conversely, suppose that V is a direct sum of the eigenspaces of T. For each i, choose an ordered basis jt of EA,. By Theorem 5.10, the union 71 U 72 U U 7fc is a basis for V. Since this basis consists of eigenvectors of T, we conclude that T is diagonalizable. Example 10 Let T be the linear operator on R1 defined by T(a,o,c,d) = (a, 6,2c, 3d). Sec. 5.2 Diagonalizability 279 It is easily seen that T is diagonalizable with eigenvalues Ai = 1, A2 = 2, and A3 = 3. Furthermore, the corresponding eigenspaces coincide with the subspaces Wi, W2, and W3 of Example 9. Thus Theorem 5.11 provides us with another proof that R4 = Wi W2 W3. EXERCISES 1. Label the following statements as true or false. (a) Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable. (b) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly dependent. (c) If A is an eigenvalue of a linear operator T, then each vector in EA is an eigenvector of T. (d) If Ai and A2 are distinct eigenvalues of a linear operator T, then EAlnEA2 = {0}. (e) Let A G M n x n (F ) and 0 = {vi,v2, ,vn} be an ordered basis for Fn consisting of eigenvectors of A. If Q is the nxn matrix whose jth column is Vj (1 < j < n), then Q~lAQ is a diagonal matrix. (f) A linear operator T on a finite-dimensional vector space is diago-nalizable if and only if the multiplicity of each eigenvalue A equals the dimension of EA-(g) Every diagonalizable linear operator on a nonzero vector space has at least one eigenvalue. The following two items relate to the optional subsection on direct sums. (h) If a vector space is the direct sum of subspaces W l5 W 2 , . . . , Nk, then Wi n Wj = {0} for 1 j. (i) ^ V = W i i=l then V = Wi W2 CE and WiDWj = {tf} for i ^ j, Wfc. 2. For each of the following matrices A G Mnxn(R), test A for diagonal-izability, and if A is diagonalizable, find an invertible matrix Q and a diagonal matrix D such that Q~lAQ D. (a) 1 2 0 1 (b) 1 3 3 1 (c) 1 4 3 2 280 Chap. 5 Diagonalization 3. For each of the following linear operators T on a vector space V, test T for diagonalizability. and if T is diagonalizable, find a basis 0 for V such that [T]a is a diagonal matrix. (a) V = P3(R) and T is defined by T(/(x)) = f'(x) + f'(x), respec-tively. (b) V = P2(R) and T is defined by T(ax2 + bx + c) = ex2 + bx + a. (c) V = R3 and T is defined by (d) V = P2(R) and T is defined by T(/(x)) = /(0) + / ( l ) (x + x2). (e) V = C2 and T is defined by T(z, w) = (z + iw, iz + w). (f) V = M2x,(/?) and T is defined by T(A) = A1. 4. Prove the matrix version of the corollary to Theorem 5.5: If A G M r i x(F) has n distinct eigenvalues, then A is diagonalizable. 5. State and prove the matrix version of Theorem 5.6. 6. (a) .Justify the test for diagonalizability and the method for diagonal-ization stated in this section, (b) Formulate the results in (a) for matrices. 7. For find an expression for A', where n is an arbitrary positive integer. 8. Suppose that A G M n X n (F) has two distinct eigenvalues, Ai and A2, and that dim(EA,) = n 1. Prove that A is diagonalizable. 9. Let T be a lineal' operator on a finite-dimensional vector space V, and suppose there exists an ordered basis i for V such that [T]^ is an upper triangular matrix. (a) Prove that the characteristic polynomial for T splits. (b) State and prove an analogous result for matrices. The converse of (a) is treated in Fxorcise 32 of Section 5.4. Sec. 5.2 Diagonalizability 281 10. Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues Ai,A2,...,Afc and corresponding multiplicities m i , m 2 , . . . , mk. Suppose that 0 is a basis for V such that [T]p is an upper triangular matrix. Prove that the diagonal entries of [T]p are Ai, A2,... .Afc and that each Ai occurs mi times (1 < i < k). 11. Let A be an n x n matrix that is similar to an upper triangular ma-trix and has the distinct eigenvalues Ai, A 2 , . . . . Afc with corresponding multiplicities mi, m 2 , . . . , mk. Prove the following statements. 12. 13. (a) tr(A) = ^ m i A i i=l (b) det(A) = (Ai)^(A2) Tn2 . . . (A/ Let T be an invertible linear operator on a finite-dimensional vector space V. (a) Recall that for any eigenvalue A of T, A - 1 is an eigenvalue of T _ 1 (Exercise 8 of Section 5.1). Prove that the eigenspace of T corre-sponding to A is the same as the eigenspace of T _ 1 corresponding to A'1. (b) Prove that if T is diagonalizable, then T _ 1 is diagonalizable. Let A G M n x n (F ) . Recall from Exercise 14 of Section 5.1 that A and A' have the same characteristic polynomial and hence share the same eigenvalues with the same multiplicities. For any eigenvalue A of A and Af, let EA and EA denote the corresponding eigenspaces for A and A*, respectively. (a) Show by way of example that for a given common eigenvalue, these two eigenspaces need not be the same. (b) Prove that for any eigenvalue A, dim(EA) = dim(EA). (c) Prove that if A is diagonalizable, then A1 is also diagonalizable. 14. Find the general solution to each system of differential equations. (a) (c) 15. Let = x + y = 3x - y (b) x'i = 8x1 + 10x2 x'2 = 5xi - 7x2 x1 = Xi x2 = x2 Xo = ^3 X3 2x3 A = fan a2i ai2 a22 0>ln 0>2n a n i a n 2 282 Chap. 5 Diagonalization be the coefficient matrix of the system of differential equations X = o-X + a i 2 x 2 + + aiTlxn x2 = a2iXi + a2 2x2 + + a2nxn x'n = a n ix i + a n 2 x 2 H h a n n x n . Suppose that A is diagonalizable and that the distinct eigenvalues of A are Alr A2,.. . , Afc. Prove that a differentiable function x: R * Rn is a solution to the system if and only if x is of the form .Ait A2t, x(t) = eAltzi + eA2t22 + + eXkizk, where Zi G Et for i = 1 ,2 , . . . , k. Use this result to prove that the set of solutions to the system is an n-dimensional real vector space. 16. Let C G Mm x n(/?), and let Y be an n x p matrix of differentiable functions. Prove (CY)' = CY', where ( r % = * for all i.j. Exercises 17 through 19 are concerned with simultaneous diagonalization. Definitions. Two linear operators T and U on a finite-dimensional vector space V are called simultaneously diagonalizable if there exists an ordered basis 0 for V such that both [T]p and [U]/? arc diagonal matrices. Similarly, A,B M n x n ( F ) are called simultaneously diagonalizable if there exists an invertible matrix Q G M n X n (F) such that both Q~l AQ and Q _ 1 BQ are diagonal matrices. 17. (a) Prove that if T and U are simultaneously diagonalizable linear operators on a finite-dimensional vector space V, then the matrices [T]^ and [U]/3 are simultaneously diagonalizable for any ordered basis 0. (b) Prove that if A and B are simultaneously diagonalizable matrices, then L.4 and LB are simultaneously diagonalizable linear operators. 18. (a) Prove that if T and U are simultaneously diagonalizable operators, then T and U commute (i.e., TU = UT). (b) Show that if A and B are simultaneously diagonalizable matrices, then A and B commute. The converses of (a) and (b) are established in Exercise 25 of Section 5.4. 19. Let T be a diagonalizable linear operator on a finite-dimensional vector space, and let m be any positive integer. Prove that T and T m are simultaneously diagonalizable. Exercises 20 through 23 are concerned with direct sums. Sec. 5.3 Matrix Limits and Markov Chains 283 20. Let Wi, W 2 , . . . , Wfc be subspaces of a finite-dimensional vector space V such that ] T w , = v. *=1 Prove that V is the direct sum of W i5 W 2 , . . . , Wfc if and only if k dirn(V) = ^ d i m ( W i ) . /=! 21. Let V be a finite-dimensional vector space with a basis 0, and let 0i,02,... ,0k be a partition of 0 (i.e., 0i,02,...,0k are subsets of 0 such that 0 = 0i U 02 U U 0k and 0{ n 0j = 0 if i ^ j). Prove that V = span(/?i) span(/?2) span(0k). 22. Let T be a linear operator on a finite-dimensional vector space V, and suppose that the distinct eigenvalues of T are Ai, A2,. - -, Afc. Prove that span({x G V: x is an eigenvector of T}) = EAX EA2 EAfc. 23. Let Wi, W2, Ki, K 2 , . . . , Kp, Mj, M 2 , . . . , M9 be subspaces of a vector space V such that W! = KiK2---Kp and W2 = MiM2-Prove that if Wx n W2 = {0}, then W i + W 2 = W i W 2 = KiK 2 Kp Mi M2 M(/. M9. 5.3* MATRIX LIMITS AND MARKOV CHAINS In this section, we apply what we have learned thus far in Chapter 5 to study the limit of a sequence of powers A, A 2 , . . . , An,..., where A is a square matrix with complex entries. Such sequences and their limits have practical applications in the natural and social sciences. We assume familiarity with limits of sequences of real numbers. The limit of a sequence of complex numbers {zm: m = 1,2, . . .} can be defined in terms of the limits of the sequences of the real and imaginary parts: If Zm rm + ism, where rm and sm are real numbers, and i is the imaginary number such that i2 = 1, then lim zm lim rm + i lim sm, m*oo m*oo moo provided that lim rm and lim sm exist. moo TOoo 284 Chap. 5 Diagonalization Definition. Let L, Ai, A2,. . . benxp matrices having complex entries. The sequence Ai, A2,. . . is said to converge to the n x p matrix L, called the limit of the sequence, if lim (Am)ij = Lij m 00 for all 1 < i < n and 1 < j < p. To designate that L is the limit of the sequence, we write lim Am = L. moo Example 1 If T^-m 1 - 3 J 3TO2 , ( 2m+l 4 J m2 + l T * I m-1 J 1 + i then lim Am = TO00 1 0 3 + 2z 0 2 e where e is the base of the natural logarithm. A simple, but important, property of matrix limits is contained in the next theorem. Note the analogy with the familiar property of limits of sequences of real numbers that asserts that if lim a m exists, then TO00 lim cam = c I lim a, TOoo m oc Theorem 5.12. Let A1.A2,... be a sequence of n x p matrices with complex entries that converges to the matrix L. Then for any P G M, x n(C) and Q G Mp x s(C), lim PAm = PL and lim AmQ = LQ. TO00 TO00 Proof. For any t (1 < % < r) and j (1 < j < p), n lim (PATn)ij= lim y^Pik(Am)kj TO'OO TOO0 * ' fc=l Sec. 5.3 Matrix Limits and Markov Chains 285 = J^Pik- lim (Am)kj = S^PikLkj = (PL)ij. ** TOQO ** fc=l fc=l Hence lim PAm = PL. The proof that lim AmQ = LQ is similar. moo TOoo Corollary. Let A G Mn x n(C) be such that lim Am = L. Then for any TO00 invertible matrix Q G MnXn(C), lim (QAQ-1)771 = QLQ-1 . Proo/. Since (QAQ-1)'1 = (QAQ~l)(QAQ~l) (QAQ~l) = QAmQ~ we have lim (QAQ~l)m = lim QA771^-1 = Q ( lim A m ) Q' 1 = QLQ TOoo TOoo 7n>oo / by applying Theorem 5.12 twice. In the discussion that follows, we frequently encounter the set 5 = { A G C : |A| < l o r A= 1}. - l Geometrically, this set consists of the complex number 1 and the interior of the unit disk (the disk of radius 1 centered at the origin). This set is of interest because if A is a complex number, then lim Am exists if and only TO00 A G S. This fact, which is obviously true if A is real, can be shown to be true for complex numbers also. The following important result gives necessary and sufficient conditions for the existence of the type of limit under consideration. Theorem 5.13. Let A be a square matrix with complex entries. Then lim Am exists if and only if both of the following conditions hold. TO00 (a) Every eigenvalue of A is contained in S. (b) If 1 is an eigenvalue of A, then the dimension of the eigenspace corre-sponding to 1 equals the multiplicity of 1 as an eigenvalue of A. One proof of this theorem, which relies on the theory of Jordan canonical forms (Section 7.2), can be found in Exercise 19 of Section 7.2. A second proof, which makes use of Schur's theorem (Theorem 6.14 of Section 6.4), can be found in the article by S. H. Friedberg and A. J. Insel, 'Convergence of matrix powers,' Int. J. Math. Educ. Sci. Technoi, 1992, Vol. 23, no. 5, pp. 765-769. 286 Chap. 5 Diagonalization The necessity of condition (a) is easily justified. For suppose that A is an eigenvalue of A such that A ^ 5. Let v be an eigenvector of A corresponding to A. Regarding v as an n x 1 matrix, we see that lim (Amv) = lim A7 noo v = Lv lim (Amn) TO0O exists, then by Theorem 5.12, where L = lim A771. But lim (Amv) TO>OC TOOC diverges because lim A771 does not exist. Hence if lim A7 TO00 TO>00 condition (a) of Theorem 5,13 must hold. Although we are unable to prove the necessity of condition (b) here, we consider an example for which this condition fails. Observe that the charac-teristic polynomial for the matrix B = is (t l)2 , and hence B has eigenvalue A = 1 with multiplicity 2. It can easily be verified that dim(EA) = 1, so that condition (b) of Theorem 5.13 is violated. A simple mathematical induction argument can be used to show that Bm = 1 m 0 1 and therefore that lim Bm does not exist. We see in Chapter 7 that if A moo is a matrix for which condition (b) fails, then A is similar to a matrix whose upper left 2 x 2 submatrix is precisely this matrix B. In most of the applications involving matrix limits, the matrix is diag-onalizable, and so condition (b) of Theorem 5.13 is automatically satisfied. In this case, Theorem 5.13 reduces to the following theorem, which can be proved using our previous results. Theorem 5.14. Let A G MnXn(C) satisfy the following two conditions. (i) Every eigenvalue of A is contained in S. (ii) A is diagonalizable. Then lim A7 m +oo exists. Proof. Since A is diagonalizable, there exists an invertible matrix Q such that Q~] AQ = D is a diagonal matrix. Suppose that D = A . o 0 A2 0 0 o 0 KJ Sec. 5.3 Matrix Limits and Markov Chains 287 Because Ai, A2 , . . . , An are the eigenvalues of A, condition (i) requires that for each *, either A* = 1 or |Ad < 1. Thus TOoo o otherwise. But since D w = / v V o 0 A271 0 0 0 the sequence D,D2,... converges to a limit L. Hence lim Am = lim (QDQ-1)7'1 = QLQ~l TO>00 TO00 by the corollary to Theorem 5.12. The technique for computing lim Am used in the proof of Theorem 5.14 TO00 can be employed in actual computations, as we now illustrate. Let A = /7 _9 _15 ( A 4 4 1 Using the methods in Sections 5.1 and 5.2, we obtain 0 0N - 0 o i such that Q~lAQ = D. Hence lim A m = lim (QDQ~l)m = lim QDmQ~1 = Q ( lim Dm) Q'1 lim TO / I 0 1 0 H r 0 0 0 d r 288 Chap. 5 Diagonalization Next, we consider an application that uses the limit of powers of a ma-trix. Suppose that the population of a certain metropolitan area remains constant but there is a continual movement of people between the city and the suburbs. Specifically, let the entries of the following matrix A represent the probabilities t hat someone living in the city or in the suburbs on January 1 will be living in each region on January 1 of the next year. Currently Currently living in living in the city the suburbs Living next year in the city Living next year in the suburbs 0.90 0.10 0.02 0.98 = A For instance, the probability that someone living in the city (on January 1) will be living in the suburbs next year (on January 1) is 0.10. Notice that since the entries of A are probabilities, they are nonnegative. Moreover, the assumption of a constant population in the metropolitan area requires that the sum of the entries of each column of A be 1. Any square matrix having these two properties (nonnegative entries and columns that sum to 1) is called a t ransi t ion matr ix or a stochastic ma-trix. For an arbitrary n x n transition matrix M, the rows and columns correspond to n s tates, and the entry M^ represents t he probability of mov-ing from state j to state i in one stage. In our example, there are two states (residing in the city and residing in the suburbs). So, for example. A2i is the probability of moving from the city to the suburbs in one stage, that is, in one year. We now determine the Suburbs Figure; 5.3 probability that a city resident will be living in the suburbs after 2 years. There are two different ways in which such a move can be made: remaining in the city for 1 year and then moving to the suburbs, or moving to the suburbs during the first year and remaining there the second year. (See Sec. 5.3 Matrix Limits and Markov Chains 289 Figure 5.3.) The probability that a city dweller remains in the city for the first year is 0.90, whereas the probability that the city dweller moves to the suburbs during the first year is 0.10. Hence the probability that a city dweller stays in the city for the first year and then moves to the suburbs during the second year is the product (0.90) (0.10). Likewise, the probability that a city dweller moves to the suburbs in the first year and remains in the suburbs during the second year is the product (0.10)(0.98). Thus the probability that a city dweller will be living in the suburbs after 2 years is the sum of these products, (0.90) (0.10) + (0.10) (0.98) = 0.188. Observe that this number is obtained by the same calculation as that which produces (A2)21, and hence (A2)2i represents the probability that a city dweller will be living in the suburbs after 2 years. In general, for any transition matrix M, the entry (Mm)ij represents the probability of moving from state j to state i in m stages. Suppose additionally that 70% of the 2000 population of the metropolitan area lived in the city and 30% lived in the suburbs. We record these data as a column vector: Proportion of city dwellers /0.70 Proportion of suburb residents 0.30 Notice that the rows of P correspond to the states of residing in the city and residing in the suburbs, respectively, and that these states are listed in the same order as the listing in the transition matrix A. Observe also that the column vector P contains nonnegative entries that sum to 1; such a vector is called a probabili ty vector. In this terminology, each column of a transition matrix is a probability vector. It is often convenient to regard the entries of a transition matrix or a probability vector as proportions or percentages instead of probabilities, as we have already done with the probability vector P. In the vector AP, the first coordinate is the sum (0.90)(0.70)+(0.02)(0.30). The first term of this sum, (0.90)(0.70), represents the proportion of the 2000 metropolitan population that remained in the city during the next year, and the second term, (0.02) (0.30), represents the proportion of the 2000 metropoli-tan population that moved into the city during the next year. Hence the first coordinate of AP represents the proportion of the metropolitan population that was living in the city in 2001. Similarly, the second coordinate of = /0.636 v0.364 represents the proportion of the metropolitan population that was living in the suburbs in 2001. This argument can be easily extended to show that the coordinates of 290 Chap. 5 Diagonalization represent the proportions of the metropolitan population that were living in each location in 2002. In general, the coordinates of AmP represent the proportion of the metropolitan population that will be living in the city and suburbs, respectively, after m stages (m years after 2000). Will the city eventually be depleted if this trend continues? In view of the preceding discussion, it is natural to define the eventual proportion of the city dwellers and suburbanites to be the first and second coordinates, respectively, of lim AmP. We now compute this limit. It is easily shown TO>00 that A is diagonalizable, and so there is an invertible matrix Q and a diagonal matrix D such that Q'1 AQ D. In fact, Q = and D = (I 0 0.88 Therefore lim A m = lim Q D m Q - J = Q [ J 1 OO TOoc U U Consequently lim AmP = LP = m*oo < r = Thus, eventually, I of the population will live in the city and | will live in the suburbs each year. Note that the vector LP satisfies A(LP) = LP. Hence LP is both a probability vector and an eigenvector of A corresponding to the eigenvalue 1. Since the eigenspace of A corresponding to the eigenvalue 1 is one-dimensional, there is only one such vector, and LP is independent of the initial choice of probability vector P. (See Exercise 15.) For example, had the 2000 metropolitan population consisted entirely of city dwellers, the limiting outcome would be the same. In analyzing the city-suburb problem, we gave probabilistic interpreta-tions of A2 and AP, showing that A2 is a transition matrix and AP is a probability vector. In fact, the product of any two transition matrices is a transition matrix, and the product of any transition matrix and probability vector is a probability vector. A proof of these facts is a simple corollary of the next theorem, which characterizes transition matrices and probability vectors. Theorem 5.15. Let M be an nxn matrix having real nonnegative entries, let v be a column vector in Rn having nonnegative coordinates, and let u G Rn be the column vector in which each coordinate equals 1. Then Sec. 5.3 Matrix Limits and Markov Chains 291 (a) M is a transition matrix if and only if Mlu = u; (b) v is a probability vector if and only if ulv = (1). Proof. Exercise. Corollary. (a) The product of two nxn transition matrices is an n x n transition matrix. In particular, any power of a transition matrix is a transition matrix. (b) The product of a transition matrix and a probability vector is a prob-ability vector. Proof. Exercise. The city-suburb problem is an example of a process in which elements of a set are each classified as being in one of several fixed states that can switch over time. In general, such a process is called a stochastic process. The switching to a particular state is described by a probability, and in general this probability depends on such factors as the state in question, the time in question, some or all of the previous states in which the object has been (including the current state), and the states that other objects are in or have been in. For instance, the object could be an American voter, and the state of the object could be his or her preference of political party; or the object could be a molecule of H 2 0 , and the states could be the three physical states in which H 2 0 can exist (solid, liquid, and gas). In these examples, all four of the factors mentioned above influence the probability that an object is in a particular state at a particular time. If, however, the probability that an object in one state changes to a differ-ent state in a fixed interval of time depends only on the two states (and not on the time, earlier states, or other factors), then the stochastic process is called a Markov process. If, in addition, the number of possible states is finite, then the Markov process is called a Markov chain. We treated the city-suburb example as a two-state Markov chain. Of course, a Markov process is usually only an idealization of reality because the probabilities involved are almost never constant over time. With this in mind, we consider another Markov chain. A certain com-munity college would like to obtain information about the likelihood that students in various categories will graduate. The school classifies a student as a sophomore or a freshman depending on the number of credits that the student has earned. Data from the school indicate that, from one fall semester to the next, 40% of the sophomores will graduate, 30% will remain sopho-mores, and 30% will quit permanently. For freshmen, the data show that 10% will graduate by next fall, 50% will become sophomores, 20% will re-main freshmen, and 20% will quit permanently. During the present year, 292 Chap. 5 Diagonalization 50% of the students at the school are sophomores and 50% are freshmen. As-suming that the trend indicated by the data continues indefinitely, the school would like to know 1. the percentage of the present students who will graduate, the percentage who will be sophomores, the percentage who will be freshmen, and the percentage who will quit school permanently by next fall; 2. the same percentages as in item 1 for the fall semester two years hence; and 3. the probability that one of its present students will eventually graduate. The preceding paragraph describes a four-state Markov chain with the following states: 1. having graduated 2. being a sophomore 3. being a freshman 4. having quit permanently. The given data provide us with the transition matrix / l 0.4 0.1 0 A = 0 0.3 0.5 0 0 0 0.2 0 lo 0.3 0.2 l) of the Markov chain. (Notice that students who have graduated or have quit permanently are assumed to remain indefinitely in those respective states. Thus a freshman who quits the school and returns during a later semester is not regarded as having changed states- the student is assumed to have remained in the state of being a freshman during the time he or she was not enrolled.) Moreover, we are told that the present distribution of students is half in each of states 2 and 3 and none in states 1 and 4. The vector P = ( o 0.5 0.5 V that describes the initial probability of being in each state is called the initial probability vector for the Markov chain. To answer question 1, we must determine the probabilities that a present student will be in each state by next fall. As we have seen, these probabilities are the coordinates of the vector AP = ( 0.4 0.1 0 0 0.3 0.5 0 0 0 0.2 0 0 0.3 0.2 l) < ) 0.5 0.5 x Sec. 5.3 Matrix Limits and Markov Chains 293 Hence by next fall, 25% of the present students will graduate, 40% will be sophomores, 10% will be freshmen, and 25% will quit the school permanently. Similarly, A2P = A(AP) provides the information needed to answer question 2: within two years 42% of the present students will graduate, 17% will be sophomores, 2% will be freshmen, and 39% will quit school. Finally, the answer to question 3 is provided by the vector LP, where L = lim Am . For the matrices / l 0.4 0.1 0 0 0.3 0.5 0 0 0 0.2 0 /0.25 0.40 0.10 v0 0.3 0.2 1/ 0.25J /0.42 0.17 0.02 [p.39 J Q = ( 19 0 0 - 7 -40 0 0 0 8 0 0 3 13 1/ and D = (I 0 0 0 0 0.3 0 0 0 0 0.2 0 o o o iy we have Q~*AQ = D. Thus L= lim Am = Q ( lim Dm) Q TO00 TO*00 / -1 / l 4 19 0 / l 0 0 0 0 - 7 -40 0 0 0 0 0 0 0 8 0 0 0 0 0 0 3 13 1/ i0 0 0 1 / a I 0 0 0 0 0 f 1 o 0 0 0 0 I 1 / I 0 0 0 4 7 1 7 0 3 7 27 56 5 7 1 8 29 56 o 0 0 1 So LP = (l f 0 0 0 0 ^ 0 56 u 0 0 0 0 29 i 56 i y / o 0.5 0.5 1 o^ f 112 0 0 53 1 1 2 / and hence the probability that one of the present students will graduate is ^ 294 Chap. 5 Diagonalization In the preceding two examples, we saw that lim AmP, where A is the TO00 transition matrix and P is the initial probability vector of the Markov chain, gives the eventual proportions in each state. In general, however, the limit of powers of a transition matrix need not exist. For example, if A/ = 0 1 1 0 then lim Mm does not exist because odd powers of M equal M and even TO00 powers of M equal / . The reason that the limit fails to exist is that con-dition (a) of Theorem 5.13 does not hold for M ( 1 is an eigenvalue). In fact. it can be shown (see Exercise 20 of Section 7.2) that the only transition matrices A such that lim Am does not exist are preeiselv those matrices for TO00 which condition (a) of Theorem 5.13 fails to hold. But even if the limit of powers of the transition matrix exists, the compu-tation of the limit may be quite difficult. (The reader is encouraged to work Exercise 6 to appreciate the truth of the last sentence.) Fortunately, there is a large and important class of transition matrices for which this limit exists and is easily computed this is the cla.ss of nyular transition matrices. Definition. A transition matrix is called regular if some power of the matrix contains only positive entries. Example 2 The transit ion matrix 0.90 0.10 0.02 0.98 of the Markov chain used in the city suburb problem is clearly regular because each entry is positive. On the other hand, the transition matrix A = of the Markov chain describing community college enrollments is not regular because the first column of .4' is / I 0 1) In 0.4 0.3 0 0.3 0.1 0.5 0.2 0.2 o 0 0 ) for any power rn. Sec. 5.3 Matrix Limits and Markov Chains Observe that a regular transition matrix may contain zero entries. For example, /0.9 0.5 0N M = 0 0.5 0.4 0.1 0 0.6, is regular because every entry of M2 is positive. The remainder of this section is devoted to proving that, for a regular transition matrix A, the limit of the sequence of powers of A exists and has identical columns. From this fact, it is easy to compute this limit. In the course of proving this result, we obtain some interesting bounds for the magnitudes of eigenvalues of any square matrix. These bounds are given in terms of the sum of the absolute values of the rows and columns of the matrix. The necessary terminology is introduced in the definitions that follow. Definitions. Let A G MnXn(C). For 1 < i, j < n, define Pi(A) to be the sum of the absolute values of the entries of row i of A, and define Vj(A) to be equal to the sum of the absolute values of the entries of column j of A. Thus n pi(A) = y jAj j - | fori = 1,2, ...n and ,(A) = ^ | A i j | for j = 1,2, i=i The row sum of A, denoted p(A), and the column sum of A, denoted f(A), are defined as p(A) = max{pi(A): 1 < i < n] and u(A) = max{vj(A): 1 < j 296 Chap. 5 Diagonalization Our next results show that the smaller of p(A) and v(A) is an upper bound for the absolute values of eigenvalues of A. In the preceding example, for instance, A has no eigenvalue with absolute value greater than 6 + /5-To obtain a geometric view of the following theorem, we introduce some terminology. For an nxn matrix A, we define the ith Gerschgorin disk C{ to be the disk in the complex plane with center An and radius ri Pi(A) An; that is, Ci = {z C: z- Au < n}. For example, consider the matrix l+2i 1 A = 2i - 3 / ' For this matrix, Ci is the disk with center 1 + 2i and radius 1, and C2 is the disk with center 3 and radius 2. (See Figure 5.4.) imaginary axis Figure 5.4 Gershgorin's disk theorem, stated below, tells us that all the eigenvalues of A are located within these two disks. In particular, we see that 0 is not an eigenvalue, and hence by Exercise 8(c) of section 5.1, A is invertible. Theorem 5.16 (Gerschgorin's Disk Theorem). Let A G MnXn(C). Then every eigenvalue of A is contained in a Gerschgorin disk. Proof. Let A be an eigenvalue of A with the corresponding eigenvector V = (vi V2 Vn) Sec. 5.3 Matrix Limits and Markov Chains 297 Then v satisfies the matrix equation Av = An, which can be written ^2 Aiivi = Xvi ( = 1 2 , . . . , n). (2) Suppose that vk is the coordinate of v having the largest absolute value; note that vk / 0 because v is an eigenvector of A. We show that A lies in Ck, that is, |A Akk < rk. For i = k, it follows from (2) that Xvk - Akkvk = ^AkjVj - Akkvk j=i J2 AkivJ Thus so ^ ^2 AkjM ^ S I-4** I K I = vk^2Akj = vkrk. vkX- Akk < vkrk; X- Au l < rk because vk > 0. Corollary 1. Let A be any eigenvalue of A G Mn X n(C). Then |A| < p(A). Proof. By Gerschgorin's disk theorem, |A Afcfcl < rk for some A;. Hence |A| = |(A - Akk) + Akk < |A - Afcfcl + |Afcfc| 298 Chap. 5 Diagonalization Corollary 3. If X is an eigenvalue of a transition matrix, then |A| < 1. The next result asserts that the upper bound in Corollary 3 is attained. Theorem 5.17. Every transition matrix has 1 as an eigenvalue. Proof. Let A be an n x n transition matrix, and let u G R' be the column vector in which each coordinate is 1. Then Alu = u by Theorem 5.15, and hence u is an eigenvector of A1 corresponding to the eigenvalue 1. But since A and A1 have the same eigenvalues, it follows that 1 is also an eigenvalue of A. I Suppose that A is a transition matrix for which some eigenvector corre-sponding to the eigenvalue 1 has only nonnegative coordinates. Then some multiple of this vector is a probability vector P as well as an eigenvector of A corresponding to eigenvalue 1. It is interesting to observe that if P is the initial probability vector of a Markov chain having A as its transition matrix, then the Markov chain is completely static. For in this situation, AmP = P for every positive integer m; hence the probability of being in each state never changes. Consider, for instance, the city-suburb problem with P = Theorem 5.18. Let A G MnXn(C) he a matrix in which each entry is positive, and let X be an eigenvalue of A such that |A| = p(A). Then X p(A) and {u} is a basis for E, where u G Cn is the column vector in which each coordinate equals 1. Proof. Let v be an eigenvector of A corresponding to A, with coordinates Vi,v2, ,vn. Suppose that vk is the coordinate of v having the largest ab-solute value, and let b vk. Then |A|6= |A||vfc| = |Anfc| = E ^ 3=1 Wb' 3=1 3=1 (3) Since |A| = p(A), the three inequalities in (3) are actually equalities; that is, (a) E Ak3V3 3=1 = ^rAkjvj, 3=1 Sec. 5.3 Matrix Limits and Markov Chains 299 (b) E l ^ i l N (c) Pk(A) = p(A). ^ | Afcj 16, and We see in Exercise 15(b) of Section 6.1 that (a) holds if and only if all the terms AkjVj (j 1 ,2 , . . . , n) are nonnegative multiples of some nonzero complex number z. Without loss of generality, we assume that z 1. Thus there exist nonnegative real numbers ci, c 2 , . . . , cn such that AkjVj = CjZ. (4) By (b) and the assumption that Akj ^ 0 for all k and j, we have |Vj | = b for j = 1 ,2 , . . . , n. Combining (4) and (5), we obtain 6 = VJ A kj A kj for j = 1,2, . . . ,n, and therefore by (4), we have Vj bz for all j. So v2 KVJ (bz bz = bzu, Kbz) and hence {u} is a basis for EA-Finally, observe that all of the entries of Au are positive because the same is true for the entries of both A and u. But Au Xu, and hence A > 0. Therefore, A = |A| = p(A). | Corollary 1. Let A G Mn X n(C) be a matrix in which each entry is positive, and let X be an eigenvalue of A such that |A| = i'(A). Then X v(A), and the dimension ofE = 1. Proof. Exercise. Corollary 2. Let A G Mnxri(C) be a transition matrix in which each entry is positive, and let X be any eigenvalue of A other than 1. Then X < 1. Moreover, the eigenspace corresponding to the eigenvalue 1 has dimension 1. Proof. Exercise. Our next result extends Corollary 2 to regular transition matrices and thus shows that regular transition matrices satisfy condition (a) of Theorems 5.13 and 5.14. 300 Chap. 5 Diagonalization Theorem 5.19. Let A be a regular transition matrix, and let X be an eigenvalue of A. Then (a) |A| < 1. (b) If X = 1, then A = 1, and dim(EA) = 1. Proof. Statement (a) was proved as Corollary 3 to Theorem 5.16. (b) Since A is regular, there exists a positive integer s such that As has only positive entries. Because A is a transition matrix and the entries of As are positive, the entries of As+1 = AS(A) are positive. Suppose that |A| = 1. Then As and As+1 are eigenvalues of As and As+1, respectively, having absolute value 1. So by Corollary 2 to Theorem 5.18, As = As+1 = 1. Thus A = 1. Let EA and E'x denote the eigenspaces of A and As, respectively, corresponding to A = 1. Then EA Q E'X and, by Corollary 2 to Theorem 5.18, dim(E'A) = 1. Hence EA = EA, and dim(EA) = 1. | Corollary. Let A be a regular transition matrix that is diagonalizable. Then lim Am exists. TO>00 The preceding corollary, which follows immediately from Theorems 5.19 and 5.14, is not the best possible result. In fact, it can be shown that if A is a regular transition matrix, then the multiplicity of 1 as an eigenvalue of A is 1. Thus, by Theorem 5.7 (p. 264), condition (b) of Theorem 5.13 is satisfied. So if A is a regular transition matrix, lim A w exists regardless of whether TO00 A is or is not diagonalizable. As with Theorem 5.13, however, the fact that the multiplicity of 1 as an eigenvalue of A is 1 cannot be proved at this time. Nevertheless, we state this result here (leaving the proof until Exercise 21 of Section 7.2) and deduce further facts about lim A771 when A is a regular m oo transition matrix. Theorem 5.20. Let A be an n x n regular transition matrix. Then (a) The multiplicity of 1 as an eigenvalue of A is 1. (b) lim A771 exists. m*oc (c) L = lim Am is a transition matrix. TO>00 (d) AL = LA = L. (e) The columns of L are identical. In fact, each column of L is equal to the unique probability vector v that is also an eigenvector of A corre-sponding to the eigenvalue 1. (f) For any probability vector w, lim (Amw) = v. moo Proof, (a) See Exercise 21 of Section 7.2. (b) This follows from (a) and Theorems 5.19 and 5.13. (c) By Theorem 5.15, we must show that ulL = u*. Now Am is a transition matrix by the corollary to Theorem 5.15, so u'L = ul lim Am = lim utAm = lim u* = u Sec. 5.3 Matrix Limits and Markov Chains 301 and it follows that L is a transition matrix, (d) By Theorem 5.12, AL = A Mm Am = mTC lim A A'1 - lim A', + ] = L. moo in.>oo Similarly, LA L. (e) Since AL L by (d), each column of L is an eigenvector of A cor-responding to the eigenvalue 1. Moreover, by (c), each column of L is a probability vector. Tims, by (a), each column of L is equal to the unique probability vector v corresponding to the eigenvalue 1 of A. (f) Let w be any probability vector, and set y = lim A'w = Lie. Then III >OC y is a probability vector by the corollary to Theorem 5.15, and also Ay ALw = Lir y by (d). Hence y is also an eigenvector corresponding to the eigenvalue 1 of A. So y v by (e). H Definition. The vector u in Theorem 5.20(c) is culled the fixed prob-ability vector or stationary vector of the regular transition matrix A. Theorem 5.20 can be used to deduce information about the eventual dis-tribution in each state of a Markov chain having a. regular transition matrix. Example 4 A survey in Persia showed that on a particular day 50% of the1 Persians preferred a loaf of bread, 30% preferred a jug of wine, and 20% preferred 'thou beside me in the wilderness.' A subsequent survey 1 month later yielded the following data: Of those who preferred a loaf of bread on the first survey. 40% continued to prefer a loaf of bread. 10%: now preferred a jug of wine, and 50% preferred 'thou'; of those who preferred a jug of wine on the first survey, 20%. now preferred a. loaf of bread, 70%: continued to prefer a jug of wine, and 10%. now preferred 'thou': of those who preferred 'thou' on the first survey, 20% now preferred a loaf of bread. 20% now preferred a. jug of wine, and 60% continued to prefer 'thou.' Assuming that this trend continues, the situation described in the preced-ing paragraph is a three-state Markov chain in which the slates are the three possible preferences. We can predict the percentage of Persians in each state for each month following the original survey. Letting the first, second, and third states be preferences for bread, wine, and 'thou', respectively, we sec that the probability vector that gives the initial probability of being in each state is P = 302 Chap. 5 Diagonalization and the transition matrix is /0.40 0.20 0.20 .4 - 0.10 0.70 0.20 . 0.50 0.10 0.60/ The probabilities of being in each state m months after the original survey arc the coordinates of the vector A'1 P. The reader may check that /0.30 /0.26 /0.252 /0.2504 AP = 0.30 , A2P = 0.32 . A*P -- 0.334 . and A4P = 0.3418 . 0.40y 0.42/ (). 111/ 0.4078/ Note the apparent convergence of A' P. Since A is regular, the long-range prediction concerning the Persians' pref-erences can be found by computing the fixed probability vector for .4. this vector is the unique probability vector v such that (A I)v = 0. Letting we see that the matrix equation (.4 I)v 0 yields the following system of linear equations: -O.GOe, + 0.20e2 + 0.20e;5 = 0 0.10v) 0.30^2 + 0.20^3 = 0 0.50i 4- 0.10e-2 - 0.40^3 - 0 . It is easilv shown that is a basis for the solution space of this system. Hence the unique fixed prob-ability vector for A is /nTTTlA 5+7+8 8 5+7+8/ Thus, in the long run. 25%. of the Persians prefer a loaf of bread. 35% prefer a jug of wine, and 40% prefer 'thou beside me in the wilderness.' Sec. 5.3 Matrix Limits and Markov Chains Note thai if - G then 303 So lim A'1 =Q 0.25 0.25 0.25 I 0.35 0.35 0.35 0.40 0.40 0.40 i o o Q- ] = Q 0 0 0 Q' 0 0 0 / Example 5 Farmers in Lamron plant one crop per year either corn, soybeans, or wheat. Because they believe in the necessity of rotating their crops, these farmers do not plant the same crop in successive years, in fact. of the total acreage on which a particular crop is planted, exactly half is planted with each of the other two crops during the succeeding year. This year, 300 acres of corn, 200 acres of soybeans, and 100 acres of wheat were planted. The situation just described is another three-state Markov chain in which the three states correspond to the planting of corn, soybeans, and wheat, respectively. In this problem, however, the amount of land devoted to each crop, rather than the percentage of the total acreage (GOO acres), is given. By converting these amounts into fractions of the total acreage, we see that the transition matrix A and the initial probability vector P of the Markov chain are A = /0 J h 0 i A 0 and P = 300 600 Jim i>(in 100 V 600 fi w The fraction of the total acreage devoted to each crop in m years is given by the coordinates of A'P. and the eventual proportions of the total acreage1 used for each crop are the coordinates of lim A' P. Thus the (wentrial 304 Chap. 5 Diagonalization amounts of land devoted to each crop are found by multiplying this limit by the total acreage; that is, the eventual amounts of land used for each crop are the coordinates of GOO lim A' P. m-oo Since A is a regular transition matrix, Theorem 5.20 shows that lim -4' III >00 is a matrix L in which each column equals the unique fixed probability vector for A. It is easily seen that the fixed probability vector for A is /i i 3 Hence L = I 3 3 :? i I i :i 3 :i 1 l i :{ 3 :5 so GOO- lim AmP = Q00LP= 200 III >TC Thus, in the long run, we expect 200 acres of each crop to be planted each year. (For a direct computation of 600 lim A'1 P. see Exercise 14.) III>OC In (his section, we have1 concentrated primarily on the theory of regular transition matrices. There is another interesting class of transition matrices that can be represented in the form I D () C where / is an identify matrix and O is a zero matrix. (Such transition ma-trices are not regular since the lower left block remains O in any power of the matrix.) The states corresponding to the identity submatrix are called absorbing states because such a. state is never left once it is entered. A Markov chain is called an absorbing Markov chain if it is possible to go from an arbitrary state into an absorbing state; in a finite number of stages. Observe that the Markov chain that describes the enrollment pattern in a community college is an absorbing Markov chain with states 1 and 4 as its ab-sorbing states. Readers interested in learning more about absorbing Markov chains are referred to Introduction to Finite. Mathematics (third edition) by Sec. 5.3 Matrix Limits and Markov Chains 305 J. Kemeuy. .]. Snell. and G. Thompson (Prentice-Hall. Inc.. Englewood Cliffs, N. J., 1974) or Discrete Mathematical Models by Fred S. Roberts (Prentice-Hall, Inc., Englewood Cliffs, N. .).. 1976). An Application In species that reproduce sexually, the characteristics of an offspring with respect to a particular genetic trait are determined by a pair of genes, one inherited from each parent. The genes for a particular trait are of two types, which are denoted by G and g. The gene (4 represents the dominant char-acteristic, and g represents the recessive characteristic. Offspring with geno-types GG or Gg exhibit the dominant characteristic, whereas offspring with genotype gg exhibit the recessive characteristic. For example, in humans, brown eyes are a dominant characteristic and blue eyes are the correspond-ing recessive characteristic; thus the offspring with genotypes GG or Gg are brown-eyed, whereas those of type gg are blue-eyed. Let us consider the probability of offspring ol' each genotype for a male parent of genotype Gg. (We assume that the population under consideration is large, that mating is random with respect to genotype, and that the distri-bution of each genotype within the population is independent of sex and life expectancy.) Let P = denote the proportion of the adult population with genotypes GG, Gg, and gg, respectively, at the start of the experiment. This experiment describes a three-state Markov chain with the following transition matrix: Genotype of female parent GG Gg Genotype GG Gg offspring ( gg 1 2 I 2I - 13. It is easily checked that I}2 contains only positive entries; so II is regular. Thus, by permitting only males of genotype Gg to reproduce, the proportion of offspring in the population having a. certain genotype will stabilize at the fixed probability vector for B. which is f w 306 Chap. 5 Diagonalization Now suppose that similar experiments are to be performed with males of genotypes GG and gg. As already mentioned, these experiments are three-state Markov chains with transition matrices A = 0 Vo o /() 0 c = o> t) respectively. In order to consider the case where all male genotypes are per-mitted to reproduce, we must, form the transition matrix M = pA + qB + rC. which is the linear combination of A. D. and C weighted by the proportion of males of each genotype. Thus (P M = ! n> 31 o V k + r 0 h b kq + r To simplify the notation, let a = p+ q and b = q + r. (The numbers a and b represent the proportions of G and g genes, respectively, in the population.) Then (a a 0 M = b a 0 b b where a + b = p + q+ r = 1. Let //, q', and /' denote the proportions of the- first-generation offspring having genotypes GG, Gg, and gg. respectively. Then = MP -/ ap+aq bp+ k + Sec. 5.3 Matrix Limits and Markov Chains where a' = p' 4- hq' and 1/ = hq' 4- r'. However 307 a' = a2 + -(2a6) = a(a + b) = a and b' - -(2a6) 4- b2 = b(a + b) = 6. Thus M = 47: so the distribution of second-generation offspring among the three genotypes is / a3+a2b I a2(a + b) (a2 M(MP) = M2P = a2b 4- ah + ah2 - ab(a 4-14-6) = 2ab ab2 + b3 ) b2(a | 6) / lr the same as the first-generation offspring. In other words. MP is the fixed probability vector for M, and genetic equilibrium is achieved in the population after only one generation. (This result is called the Hardy Weinberg law.) Notice that in the important special case; that a b (or equivalently, that p r), the distribution at equilibrium is MP - 2ab I rt EXERCISES 1. Label the following statements as true or false (a) UAe 'nx n {(.) and lim A' = L. then, for any invertible matrix (b) (c) Q e MnXn(C), we have lim QAmQ~i = QLQ If 2 is an eigenvalue of A 1 ii x n (C). then lim A' does not exist. Anv vector (d) (e) l'2 such that .r + x-2 4- 4- xn = 1 is a probability vector. The sum of the entries of each row of a transition matrix equals 1. The product of a transition matrix and a probability vector is a probability vector. 308 Chap. 5 Diagonalization (f) Let z be any complex number such that z < 1. Then the matrix does not have 3 as an eigenvalue, (g) Every transition matrix has 1 as an eigenvalue. (h) No transition matrix can have 1 as an eigenvalue, (i) If A is a transition matrix, then lim Am exists. m>oo (j) If A is a regular transition matrix, then lim Am exists and has m+oo rank 1. 2. Determine whether lim Am exists for each of the following matrices moo A, and compute the limit if it exists. (a) (d) 0.1 0.7 0.7 0.1 -1 .8 -0 .8 4.8 2.2 (b) (e) -1 .4 -2.4 0.8 1.8 - 2 - 1 4 3 (c) (f) 0.4 0.7 0.6 0.3 2.0 -0.5 3.0 -0.5 G) 3 - 7 4- 2i 3 - 1 3 4-62 3 - 5 4- i 3 - 5 4-Gi G 6 7 - 2z 35 - 20?: 3. Prove that if A, A2, . is a sequence of n x p matrices with complex entries such that lim Am = L. then lim (AmY = V. m oo m>oo 4. Prove that if A MnXU(C) is diagonalizable and L = lim Am exists, then either L = In or rank(L) < n. Sec. 5.3 Matrix Limits and Markov Chains 309 5. Find 2 x 2 matrices A and B having real entries such that lim Am, moo lim Bm. and lim (AB)m all exist, but m-oc in ->oc lim (AB)m ^ ( lim Am)( lim Bm). 6. A hospital trauma unit has determined that 30% of its patients are ambulatory and 70% are bedridden at the time of arrival at the hospital. A month after arrival, 60% of the ambulatory patients have recovered, 20% remain ambulatory, and 20% have become bedridden. After the same amount of time, 10% of the bedridden patients have recovered, 20% have become ambulatory, 50%: remain bedridden, and 20% have died. Determine the percentages of patients who have recovered, are ambulatory, are bedridden, and have died 1 month after arrival. Also determine the eventual percentages of patients of each type. 7. A player begins a. game of chance by placing a marker in box 2, marked Start. (See Figure 5.5.) A die is rolled, and the marker is moved one square to the left if a I or a 2 is rolled and one square to the right if a 3. 4. 5. or G is rolled. This process continues until the marker lands in square 1. in which case the player wins the game, or in square 4, in which case the player loses the game. What is the probability of winning this game? Hint: Instead of diagonali/ing the appropriate transition matrix Win 1 Start 2 3 Lose 4 Figure 5.5 ,4, it is easier to represent e310 Chap. 5 Diagonalization (g) / 0 0 0 A o o o i i o I o i (h) 0 0 V1 4 o o 9. Compute lim A7' if it exists, for each matrix A in Exercise 8. moc 10. Each of the matrices that follow is a regular transition matrix for a three-state Markov chain. In all cases, the initial probability vector is 11. 12. For each transition matrix, compute the proportions of objects in each state after two stages and the eventual proportions of objects in each state by determining the fixed probability vector. /0.6 0.1 0 . l /0.8 0.1 0.2 /0.9 0.1 ().l (a) 0.1 0.9 0.2 (b) 0.1 0.8 0.2 (c) 0.1 0.6 0.1 0.3 0 0.7/ V0.1 0.1 0.6/ 0 0.3 0.8/ (d) 0.4 0.1 0.5 0.2 0.7 0.1 0.2 0.2 0.6 (e) 0.5 0.3 0.2 0.5 0.3 0.2 0.2 0.3 0.5 (f) 0.6 0 0.4 0.2 0.8 0.2 0.2 0.2 0.4 In 1940, a county land-use survey showed that 10% of the county land was urban, 50%i was unused, and 40% was agricultural. Five years later. a follow-up survey revealed that 70%. of the urban land had remained urban, 10% had become unused, and 20% had become agricultural. Likewise, 20% of the unused land had become urban. 60% had remained unused, and 20% had become agricultural. Finally, the 1945 survey showed that 20% of the agricultural land had become unused while 80% remained agricultural. Assuming that the trends indicated by the 1945 survey continue, compute the percentages of urban, unused, and agricultural land in the county in 1950 and the corresponding eventual percentages. A diaper liner is placed in each diaper worn by a baby. If. after a diaper change, the liner is soiled, then it is discarded and replaced by a new liner. Otherwise, the liner is washed with the diapers and reused, except that each liner is discarded and replaced after its third use (even if it has never been soiled). The probability that the baby will soil any diaper liner is one-third. If there are only new diaper liners at first, eventually what proportions of the diaper liners being used will be new, Sec. 5.3 latrix Limits and Markov Chains 311 once used, and twice used? Hint: Assume that a diaper liner ready for use is in one of three states: new. once used, and twice used. After its use. it then transforms into one of the three states described. 13. In 1975. the automobile industry determined that 40% of American car owners drove Large cars, 20% drove intermediate-sized cars, and 40% drove small cars. A second survey in 1985 showed that 70% of the large-car owners in 1975 still owned large cars in 1985, but 30% had changed to an intermediate-sized car. Of those who owned intermediate-sized cars in 1975, 10% had switched to large cars. 70% continued to drive intermediate-sized cars, and 20% had changed to small cars in 1985. Finally, of the small-car owners in 1975, 10% owned intermediate-sized cars and 90% owned sma.ll cars in 1985. Assuming that these trends continue, determine the percentages of Americans who own cars of each size in 1995 and the corresponding eventual percentages. 14. Show that if A and P are as in Example 5, then ( rm rm I 1 'm+1 >'m + i rin rm- i J . rm I 1 7'm-l !' Vm = 3 om-Deduce that /30(r 600(4' P) - A' 200 ] V 100 200 200 2'> 200 ( -1 ) ' ' 1 1 2' IOO': (100) 15. Prove that if a 1-dimensional subspace W of R' contains a nonzero vec-tor with all nonnegative entries, then W contains a unique probability vector. 16. Prove Theorem 5.15 and its corollary. 17. Prove the two corollaries of Theorem 5.18. 18. Prove the corollary of Theorem 5.19. 19. Suppose that M and M' are n x n transition matrices. 312 Chap. 5 Diagonalization (a) Prove that if M is regular. N is any n x n transition matrix, and c is a real number such that 0 < c < 1, then cM + (1 c)N is a regular transition matrix. (b) Suppose that for all i. j, we have that M[- > 0 whenever A/;,- > 0. Prove that there exists a transition matrix N and a real number c with 0 < c. < 1 such that M'= cM 4- (1 - c)JV. (c) Deduce that if the nonzero entries of M and M' occur in the same positions, then M is regular if and only if M' is regular. The following definition is used in Exercises 20-24. Definition. For A 6 Mn x(C), define e = lim Bm. where III- ' oc A2 Am Bm - I 4 A 2! (see Exercise 22). Thus e is the .sum of the infinite series 1 + A + A*_ 3! and Bm is the mth partial sum of this series. (Note the analogy with the power series c' = 1 2! a:' 4! which is valid for all complex numbers a.) 20. Compute c and c1. where () and / denote the n x n zero and identity matrices, respectively. 21. Let P~lAP = D be a diagonal matrix. Prove that eA = PeDP~1. 22. Let A G Mnxn(C) be diagonalizable. Use the result of Exercise 21 to show that eA exists. (Exercise 21 of Section 7.2 shows that e exists for every Ac MX(C).) 23. Find A,B e M2x2(/?) such that eAeB ^ eA+B. 24. Prove that a differentiable function x: B -^ R' is a solution to the system of differential equations defined in Exercise 15 of Section 5.2 if and only if x(t) = e v for some u R', where A is defined in that exercise. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 313 5.4 INVARIANT SUBSPACES AND THE CAYLEY-HAMILTON THEOREM In Section 5.1. we observed that if v is an eigenvector of a linear operator T. then T maps the span of {r} into itself. Subspaces that are mapped into themselves are of great importance in the study of linear operators (sec. e.g.. Exercises 28 42 of Section 2.1). Definition. Let T be a linear operator on a vector space V. ,4 suhspace W of/ is called a T-invariant subspace of V if T(W) C_ W. thai is. if T(v) W for all v 6 W. Example 1 Suppose that T is a linear operator on a vector space V. Then the following subspaces of V arc- T-invariant: 1. {0} 2. V 3. R(T) f. N(T) 5. E.- for any eigenvalue A of T. The proofs that these subspaces are T-invariant are left as exercises. (Sec Exercise 3.) Example 2 Let T be the linear operator on R'{ defined by T(a,6,c) = (a + b,b + c,0). Then the .///-plane = {(.r.vy.O): x.y G R) and the .r-axis - {(./:, 0.0): x E R} are T-invariant subspaces of R5. Let T be a linear operator on a vector space V, and let. ;/: be a nonzero vector in V. The? subspace W-span({.r.T(:r).T2(.70,...}) is called the T-cyclic subspace of V generated by ./'. It is a. simple matter to show that W is T-invariant.. In fact, W is the 'smallest' T-invariant sub-space of V containing x. That is. any T-invariant subspace of V containing x must also contain W (see Exercise 11). Cyclic subspaces have various uses. We apply them in this section to establish the Cayley -Hamilton theorem. In Exercise 41. we outline a method for using cyclic subspaces to compute the characteristic polynomial of a linear operator without resorting to determi-nants. Cyclic subspaces also play an important role in Chapter 7, when1 we study matrix representations of nondiagonalizable linear operators. 314 Chap. 5 Diagonalization Example 3 Let T be the linear operator on R,{ denned by T(a, byc) = (-b + c, a + c, 3c). We determine the T-cyclic subspace generated by e. = (1.0,0). Since T ( e i ) = T ( l , 0 , 0 ) = (0,l ,0) = e2 and T2(ex) = T(T(ei)) - T(e2) = (-1,0,0) = - c , . it follows that span({ei,T(e1),T2(e1),. . .}) = span({ei,e2}) - {(a.t.O): s.t e R}. Example 4 Let T be the linear operator on P(R) defined by T(f(x)) = f'(x). Then the T-cyclic subspace generated by x is span({x , 2./:. 2}) = P-2(R). The existence of a T-invariant subspace provides the opportunity to define a new linear operator whose domain is this subspace. If T is a linear operator on V and W is a T-invariant subspace of V, then the restriction Tw of T to W (sec Appendix B) is a mapping from W to W. and it follows that Tw is a linear operator on W (see Exercise 7). As a linear operator, Tw inherits certain properties from its parent operator T. The following result illustrates one way in which the two operators are linked. Theorem 5.21. Let T be a linear operator on a finite-dimensional vector space V, and let W be a T-invariant subspace of V. Then the characteristic polynomial of Tw divides the characteristic polynomial of T. Proof. Choose an ordered basis 7 = {v ,t>2, . , Vk] for W, and extend it to an ordered basis (3 {v. ??2, Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 315 Example 5 Let T be the linear operator on R' defined by T(a, 6, c, d) = (a 4-6 4- 2c - d, b + d, 2c - d. c + d), and let W - {(/. .s. 0, 0): t. s 6 R}. Observe that W is a T-invariant subspace of R' because, for any vector (a,6,0,0) G R4, T(.6.0.0) - (a + b. 6,0,0) W. Lei -) {> .). which is an ordered basis for W. Extend 7 to the standard ordered basis 0 for R1. Then R> - [Twl-v = 4 A=[T]f3 = ( 1 2 - 1 0 1 0 1 0 0 2 - 1 o 0 1 ) in the notation of Theorem 5.21. Let f(t) be the characteristic polynomial of T and g(t) be the characteristic polynomial of Tw- Then f(t) = det(A - tl4) = det A - i 1 2 - 1 0 1 - /. 0 1 0 0 2-t - 1 0 0 1 1 - tj ' ' o ! ! . - , 2 r ( r _ - g(t)- det 2 / - < In view of Theorem 5.21. we may use the characteristic polynomial of Tw to gain information about the characteristic polynomial of T itself. In this re-gard, cyclic subspaces are useful because the characteristic polynomial of the restriction of a linear operator T to a cyclic subspace is readily computable. Theorem 5.22. Let T be a linear operator on a finite-dimensional vector space V. and let W denote the T-cyclic subspace ofV generated by a nonzero vector v 6 V. Let k = dim(W). Then (a) {v,T(v),T2(v),... .Tk '(')} ^ a basis for W. (b) Ifaov + aiT(v) + Hafc_iTfc' 1(v)+Tk(v) = 0, then the characteristic polynomial of Tw is f(t) = (1) (an 4- a.t + + ak-it A- 1 316 Chap. 5 Diagonalization Proof, (a) Since v ^ 0, the set {v} is linearly independent. Let j be the largest positive integer for which 0 = {v,T(v),...,V-1(v)} is linearly independent. Such a j must exist because V is finite-dimensional. Let Z = span(.tf). Then ji is a basis for Z. Furthermore, T^(v) 6 Z by Theorem 1.7 (p. 39). We use this information to show that Z is a T-invariant subspace of V. Let. w E Z. Since w is a linear combination of the vectors of (3, there exist scalars bo,b.... ,bj-^ such that w = bnv + 6i T(i bj-iV-Hv), and hence T(w) = b0T(v) + b{T2{v) 4- 4- fc-iT'(v). Thus T(w) is a linear combination of vectors in Z, and hence belongs to Z. So Z is T-invariant. Furthermore, e Z. By Exercise 11. W is the smallest T-invariant subspace of V that contains v. so that. W C Z. Clearly. Z C W. and so we conclude that Z = W. It follows that d is a basis for W. and therefore dim(W) = j. Thus j = k. This proves (a). (b) Now view B (from (a)) as an ordered basis for W. Let a,). u..... a,k be the scalars such that k I a0v + a{T(y) + + ak-,Tk J(v) + Th(v) = 0. Observe that /() 0 I 0 an - i [Tw]/s = 0 0 1 -ak-J which has the characteristic polynomial / ( i ) = (-l) f c(a0-r-a1 i4----4-a f c_1 i f c-1 tk) by Exercise 19. Thus f(t) is the characteristic polynomial of Tw, proving (b). Example 6 Let T be the linear operator of Example 3, and let W = span({ei,e2}), the T-cyclic subspace generated by e. We compute the characteristic polyno-mial f(t) of Tw in two ways: by means of Theorem 5.22 and by means of determinants. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 317 (a) By means of Theorem, 5.22. From Example 3, we have that {ei.e^} is a cycle that generates W, and that T2(ci) = e. Hence ex+{)T(e,) + T2(cl)^(). Therefore, by Theorem 5.22(b), f(t) = (-l)2(l + 0t + t2) = t2 + l. (b) By means of determinants. Let ft = {ci,e2j, which is an ordered basis for W. Since T(ei) = c> and T(e2) = e., we have [Tw],, - 0 - 1 and therefore. 1 -t] The Cayley-Hamilton Theorem As an illustration of the importance of Theorem 5.22, we prove a well-known result that is used in Chapter 7. The reader should refer to Ap-pendix E for the definition of /(T), where T is a linear operator and f(x) is a polynomial. Theorem 5.23 (Cayley-Hamilton). Let T be a linear operator on a finite-dimensional vector space V. and let f(t) be the characteristic polyno-mial of T. Then / (T) = T0. the zero transformation. That is, T 'satisfies' its characteristic equation. Proof. We show that /(T)(v) = 0 for all v E V. This is obvious if v = 0 because / (T) is linear: so suppose that v ^ 0. Let W be the T-cyclic subspace generated by u, and suppose that dim(W) = k. By Theorem 5.22(a), there exist scalars an, , dk l .such that -A--1 v)+Tk(v) = 0. a0v + aiT(v) + + dk-iV Hence Theorem 5.22(b) implies that g{t) = (- l) f c(a0 4- ait +- + afc-it*'1 4- tk) is the characteristic polynomial of Tw- Combining these two equations yields 9(T)(v) = (-l) f c(a0 l + oiT + + afc-iT^'1 + Tk)(v) = 0. By Theorem 5.21. g(t) divides f(i); hence there exists a polynomial q(t) such that f(t) = q(t)g(t). So f(T)(v) = q(T)g(T)(v) = q(T)(g(T)(v)) = q(J){0) = 0. 318 Chap. 5 Diagonalization Example 7 Let T be the linear operator on R2 defined by T(a, b) = (a + 2b. 2a -1- b). and let (3= {ei,e2}. Then A = 1 2 - 2 1/ ' where A = [T]^. The characteristic polynomial of T is, therefore. /(f) = det(,4 - / /) - det f ^ t % ) = t2 - 2t 4- 5. It is easily verified that T0 = / (T) = T2 - 2T + 51. Similarly. - 3 4 , (-2 - 4 / ', f(A) = A2-2A + 5/ = 0 0 0 ( ) / ' 4 - 3 4 - 2 0 5 Example 7 suggests the following result. Corollary (Cayley-Hamilton Theorem for Matrices) . Let A be an n x n matrix, and let f(t) be the characteristic polynomial of A. Then f(A) = O, the n x n zero matrix. Proof. See Exercise 15. 1 Invariant Subspaces and Direct Sums* It is useful to decompose a finite-dimensional vector space V into a direct sum of as many T-invariant subspaces as possible because the behavior of T on V can be inferred from its behavior on the direct summands. For example. T is diagonalizable if and only if V can be decomposed into a direct sum of one-dimensional T-invariant subspaces (see Exercise 36). In Chapter 7. we consider alternate ways of decomposing V into direct sums of T-invariant subspaces if T is not diagonalizable. Wre proceed to gather a few facts about direct sums of T-invariant subspaces that are used in Section 7.4. The first of these facts is about characteristic polynomials. Theorem 5.24. Let T be a linear operator on a finite-dimensional vector space V, and suppose that V = W| W-2 Nk, where W, is a T-invariant subspace of V for each i (1 < i < k). Suppose that /,(/) is the characteristic polynomial of Tw, (1 < i < k). Then fi(t)'f2(t) ,//,-(/) is the characteristic polynomial of T. 'This subsection uses optional material on direct sums from Section 5.2. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 319 Proof. The proof is by mathematical induction on k. In what follows. f(l) denotes the characteristic polynomial of T. Suppose first that k 2. Let ;3 be an ordered basis for Wj, fa an ordered basis for W2, and 0 = fa U 02-Then 3 is an ordered basis for V by Theorem 5.10(d) (p. 276). Let A = [T^, B = [TwJ/Si) a n d J^ 2 = [Tw2]/V By Exercise 34, it follows that A = Bi O O' B2y ' where O and O' are zero matrices of the appropriate sizes. Then / ( / ) = det(i4 - tl) = det(B, - /) det(B2 - tl) = fi(t)-f2(t) as in the proof of Theorem 5.21. proving the result for k 2. Now assume t hat the theorem is valid for k summands, where k 1 > 2, and suppose that V is a direct, sum of k subspaces, say. v = w , w2].: e w f e . Let W = Wi I- W2-l hWfc 1. It is easily verified that W is T-invariant and that V - W I Wfc. So by the case for k = 2, / ( / ) - g(t)-fk(t), where g(t) is the characteristic polynomial of Tw- Clearly W = Wi W2- -Wfc_i, and therefore g(t) fi(t)-f2(t) fk-i(t) by the induction hypothesis. We conclude that f{t)=g(t)'fk(t) - fi(t)-f2{t) fk(t). I As an illustration of this result, suppose that T is a diagonalizable lin-ear operator on a finite-dimensional vector space V with distinct eigenvalues A]. A2 A/,.. By Theorem 5.11 (p. 278), V is a direct sum of the eigenspaces of T. Since each eigenspace is T-invariant, we may view this situation in the context of Theorem 5.24. For each eigenvalue A,;, the restriction of T to E, has characteristic polynomial (A; f)'1' where m., is the dimension of E^.. By Theorem 5.24. the characteristic polynomial f(t) of T is the product /(*) = ( A i - * ) m i ( A 2 - t ) m a ' - ( A f c - t ) m * . It follows that the multiplicity of each eigenvalue is equal to the dimension of the corresponding eigenspace, as expected. Example 8 Lei T be the linear operator on R4 denned by T(a, b, c, d) = (2a - b, a b. c - d, c + d), and let W, - {(s.t.().()): s.t E R] and W2 - {(0,0,s,t): s,t E R}. Notice that W] and W2 are each T-invariant and that R1 = Wi W2. Let fa = {ei,e2}, 02 = {e3,e4}, and 0 = fa U ft2 {ei,e2,e3,e4}. Then fa is an 320 Chap. 5 Diagonalization ordered basis for Wi, fa is an ordered basis for VV2, and 0 is an ordered basis for R4. Let A = [T]a, Bi = [Twjfr, and B2 = [Tw 2k- Then B, = 2 - 1 Bo = - 1 1 and (2 - 1 0 0 A = O Bo 1 0 0 0 0 I o 0 1 7 Let ./(f), / i ( i ) , and f2(t) denote the characteristic polynomials of T. Tw,-and Tw2r respectively. Then f(t) = det(A - tl) = det(5i - tI)>det(B2 - II) = / , (t)-f2(t). The matrix A in Example 8 can be obtained by joining the mat rices B and B2 in the manner explained in the next definition. Definition. Let # , E UmXIII{F), and let B2 E Mxn(F). We define the direct sum of B and B2) denoted B OjB2, as the (m + ri) x (rn + n) matrix A such that ({Bxlij fori Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 321 Then Bi B> ' lh -( 1 2 1 1 0 0 0 0 0 0 3 0 0 0 0 ^ 0 0 0 0 0 0 1 1 1 0 0 ^ 0 0 0 0 2 1 2 3 1 1 J The final result of this section relates direct sums of matrices to direct sums of invariant subspaces. It is an extension of Exercise 34 to the case k>2. Theorem 5.25. Let T be a linear operator on a finite-dimensional vector space V. and let Ni,N2,... .Nk be T-invariant subspaces of V such that V - Wi : WV! I WA.. For each i, let .4, be an ordered basis for W;, and let 3 = 0} U 02 U U 0k. Let A = [T]0 and B, = [Tw ,c for i = 1 ,2 , . . . , k. Then A = BX B2 Bk. Proof. See Exercise 35. EXERCISES 1. Label the following statements as true or false. (a) There exists a linear operator T with no T-invariant subspace. (b) If T is a linear operator on a finite-dimensional vector space V and W is a T-invariant subspace of V. then the characteristic polyno-mial of Tw divides the characteristic polynomial of T. (c) Let T be a linear operator on a finite-dimensional vector space V, and let v and w be in V. If W is the T-cyclic subspace generated by v. W is the T-cyclic subspace generated by w. and W W . then v iv. (d) If T is a linear operator on a finite-dimensional vector space V. then for any r E V the T-cyclic subspace generated by v is the same as the T-cyclic subspace generated by T(r). (e) Let T be a linear operator on an //-dimensional vector space. Then there exists a polynomial g(t) of degree // such that g(T) = T0. (f) Any polynomial of degree // with leading coefficient (1)' is the characteristic polynomial of some linear operator. (g) If T is a linear operator on a finite-dimensional vector space V. and if V is the direct sum of/,- T-inva.ria.nt subspaces, then there is an ordered basis 0 for V such that [Tj:; is a direct sum of k matrices. 322 Chap. 5 Diagonalization 2. For each of the following linear operators T on the vector space V. determine whether the given subspace W is a T-invariant subspace of V. (a) V = P3(R), T(/(a:)) - fix), and W = P2(R) (b) V - P(R), T(f(:r)) =- xf(x), and W = P2(R) (c) V - R;i, T(a, b, c) = (a 4- b + c, a 4-6 4- c, a + 6 + c). and W = {{Lt.t): te R (d) V = C([0, 1]), T(/(/)) = [./;,' f(x)dx] I. and W = { / E V: f(l) = ai 4- b for some a and 6} '0 1' (e) V = M2x2(R), T(A) = A, and W = {4 G V: A* = A) 3. Let T be a linear operator on a finite-dimensional vector space V. Prove that the following subspaces are T-invariant. (a) {0}andV (b) N(T) and R(T) (c) E, for any eigenvalue A of T 4. Let T be a linear operator on a vector space V, and let W be a T-invariant subspace of V. Prove that W is p(T)-invariant for any poly-nomial g(t). 5. Let T be a linear operator on a vector space V. Prove that the inter-section of any collection of T-invariant subspaces of V is a T-invariant subspace of V. 6. For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z. (a) V = R4, T(tt, b. c, d) = {a + b, b - c, a + c. a 4- d), and z = cx. (b) V - P3(i?.), T(/(s)) = /'(./). and z (c) V = IV!L. (d) v = .3 2(R), T(A) = A' and z = 0 1 1 0 ,(R), T(.4) - Q 2) A a n d z = 7. Prove that the restriction of a linear operator T to a. T-invariant sub-space is a linear operator on that subspace. 8. Let T be a linear operator on a vector space with a T-invariant. subspace W. Prove that if v is an eigenvector of Tw with corresponding eigenvalue A, then the same is true for T. 9. For each linear operator T and cyclic subspace W in Exerc4.se 6, compute the characteristic polynomial of Tw in two ways, as in Example 6. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 323 10. For each linear operator in Exercise 6, find the characteristic polynomial f(t) of T, and verify that the characteristic polynomial of Tw (computed in Exercise 9) divides f(t). 11. Let T be a linear operator on a vector space V, let u b e a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. Prove that (a) W is T-invariant. (b) Any T-invariant subspace of V containing v also contains W. 12. Prove that A = Bx Bi O B, in the proof of Theorem 5.21. 13. Let T be a linear operator on a vector space V, let v be a nonzero vector in V. and let W be the T-cyclic subspace of V generated by v. For any w E V, prove that w E W if and only if there exists a polynomial g(t) such that w = g(T)(?;). 14. Prove that the polynomial g(i) of Exercise 13 can always be chosen so that its degree is less than dim(W). 15. Use the Cayley-Hamilton theorem (Theorem 5.23) to prove its corol-lary for matrices. Warning: If f(t) = det (.A tl) is the characteristic polynomial of A, it is tempting to 'prove' that f(A) = O by saying uf(A) = det(A - AI) = det(O) = 0.' But this argument is nonsense. Why? 16. Let T be a linear operator on a finite-dimensional vector space V. (a) Prove that if the characteristic polynomial of T splits, then so does the characteristic polynomial of the restriction of T to any T-invariant subspace of V. (b) Deduce that if the characteristic polynomial of T splits, then any nontrivial T-invariant subspace of V contains an eigenvector of T. 17. Let A be an n x n matrix. Prove that dim(span({/n, A,A2 , .--})) < n. 18. Let A be an n x n matrix with characteristic polynomial f(t) = (-l)ntn + an-it n- l + ait + OQ. (a) Prove that A is invertible if and only if an ^ 0. (b) Prove that if A is invertible, then A'1 = ( - l / a o ) [ ( - l ) M n An 1 n ,An~2 ai J. Un 324 Chap. 5 Diagonalization (c) Use (b) to compute A ! for 19. Let A denote the k X k matrix /() 0 0 - a 1 0 0 -a.! 0 1 0 -o.2 0 0 0 0 0 -ak-2 1 -Ok-iJ where ao,ai, , ak- are arbitrary scalars. Prove that the character-istic polynomial of A is (-!)*(). 21. Let T be a linear operator on a two-dimensional vector space V. Prove that either V is a T-cyclic subspace of itself or T = cl for some scalar c. 22. Let T be a linear operator on a two-dimensional vector space V and suppose that T ^ c for any scalar c. Show that if U is any linear operator on V such that UT = TU, then U = q(T) for some polvnoniial g(t). 23. Let T be a linear operator on a finite-dimensional vector space V, and let W be a T-invariant subspace of V. Suppose that Vi,v2, vk are eigenvectors of T corresponding to distinct eigenvalues. Prove that if v 4- v2 4 (- vk is in W, then vi E W for all i. Hint: Use mathematical induction on k. 24. Prove that the restriction of a diagonalizable linear operator T to any nontrivial T-invariant subspace is also diagonalizable. Hint: Use the result of Exercise 23. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 325 25. (a) Prove the converse to Exercise 18(a) of Section 5.2: If T and U are diagonalizable linear operators on a finite-dimensional vector space V such that UT = TU, then T and U are simultaneously diagonalizable. (See the definitions in the exercises of Section 5.2.) Hint: For any eigenvalue A of T, show that E is U-invariant, and apply Exercise 24 to obtain a basis for EA of eigenvectors of U. (b) State and prove a matrix version of (a). 26. Let T be a linear operator on an //-dimensional vector space V such that T has n distinct eigenvalues. Prove that V is a T-cyclic subspace of itself. Hint: Use Exercise 23 to find a vector v such that {v, T(v),..., T n _ 1 (v)} is linearly independent. Exercises 27 through 32 require familiarity with quotient spaces as defined in Exercise 31 of Section 1.3. Before attempting these exercises, the reader should first review the other exercises treating quotient spaces: Exercise 35 of Section 1.6, Exercise 40 of Section 2.1, and Exercise 24 of Section 2.4. For the purposes of Exercises 27 through 32, T is a fixed linear operator on a finite-dimensional vector space V, and W is a nonzero T-invariant subspace of V. We require the following definition. Definition. Let T be a linear operator on a vector space V, and let W be a T-invariant subspace of V. Define T: V/W -* V/W by T(v + W) = T(v) 4- W for any v 4- W E V/W. 27. (a) Prove; that T is well defined. That. is. show that T(e + W) = T(v' + W) whenever v + N = v' + W. (b) Prove that T is a linear operator on V/W. (c) bet //: V > V/W be the linear transformation defined in Exer-cise 40 of Section 2.1 by TJ(V) = v + W. Show that the diagram of Figure 5.6 commutes; that is, prove that rjT = Tip (This exercise does not require the assumption that V is finite-dimensional.) V V V/W V/W Figure 5.6 28. Let f(t). g(t), and h(t) be the characteristic polynomials of T, Tw. and T, respectively. Prove that f(t) = g(i)h(i). Hint: Extend an ordered basis 7 = {v,v2,... ,vk} for W to an ordered basis ft = {vi, v2,..., vk, Vfc+ij ,vn} for V. Then show that the collection of 326 Chap. 5 Diagonalization cosets a = {vk+i 4- W, vk .2 4- W , . . . , v + W} is an ordered basis for V/W, and prove; that where j = [T]7 and S 3 = [T]f,. 29. Use the hint in Exercise 28 to prove that if T is diagonalizable. then so isT. 30. Prove that if both Tw and T are diagonalizable and have no common eigenvalues, then T is diagonalizable. The results of Theorem 5.22 and Exercise 28 are useful in devising methods for computing characteristic polynomials without the use of determinants. This is illustrated in the next exercise. / I , - 3 31. Let A = 2 3 4 . let, T = LA. and let W be the cyclic subspace V 2 l) of R generated by e. (a) Use Theorem 5.22 to compute1 the characteristic polynomial of Tw. (b) Show that {e2 4 W} is a basis for R;i/W, and use this fact to compute the characteristic polynomial of T. (c) Use the results of (a) and (b) to find the characteristic polynomial of A. 32. Prove the converse to Exercise 9(a) of Section 5.2: If the characteristic polynomial of T splits, then there is an ordered basis 0 for V such that [T].j is an upper triangular matrix. Hints: Apply mathematical induction to diin(V). First prove that T has an eigenvector v. 1(4 W = span({e}), and apply the induction hypothesis to T: V/W * V/W. Exercise 35(b) of Section 1.6 is helpful here. Exercises 33 through 40 arc; concerned with direct sums. 33. Let T be a linear operator on a vector space V. and let Wi. W> W/, be; T-invariant subspaces of V. Prove that W, 4- W2 4 h Wfc is also a T-invariant subspace of V. 34. Give a direct proof of Theorem 5.25 for the case k = 2. (This result is used in the proof of Theorem 5.24.) 35. Prove Theorem 5.25. Hint: Begin with Exercise 34 and extend it using mathematical induction on k.. the number of subspaces. Sec. 5.4 Invariant Subspaces and the Cayley-Hamilton Theorem 327 36. Lei T be a linear operator on a finite-dimensional vector space V. Prove that T is diagonalizable if and only if V is the direct sum of one-dimensional T-invariant subspaces. 37. Let T be a linear operator on a. finite-dimensional vector space1 V. and let W|.W_> WA. be T-invariant subspaces of V such that V = WiO W2 ] DWfc. Prove that det(T) =det(TW |)let(Tw,) 38. Let T be a linear operator on a finite-dimensional vector space V. and let Wi,W-_> W,, be T-invariant subspaces of V such that V W| | Wj : W/,. Prove that T is diagonalizable if and only if Tw , is diagonalizable for all /'. 39. Let C be a collection of diagonalizable linear operators on a finite-dimensional vector space V. Prove that there is an ordered basis .3 such that [T].( is a diagonal matrix for all T E C if and only if the; operators of C commute under composition. (This is an extension of Exercise 25.) Hints for the ease that the operators commute: The result is trivial if each operator has only OIK1 eigenvalue. Otherwise, establish the general result by mathematical induction on dim(V), using the fact that V is the direct sum of the eigenspaces of some operator in C that. has more than one eigenvalue. 40. Let B. B2..... B, be square matrices with entries in the same field, and let A B B> > i- Bk. Prove that the characteristic polynomial of A is the product of the characteristic polynomials of the P,'s. 41. Let A = 2 n 2 n2 - n. + I ir n 4- 2 7 Find the characteristic polynomial of A. Hint: First prove that A has rank 2 and that span({(l, 1 1), (1,2 //)}) is L^-invariant. 42. Let .4 E Mx(/i) be the matrix defined by ,4,;, 1 for all i and j. find the characteristic polynomial of .4. 328 Chap. 5 Diagonalization INDEX OF DEFINITIONS FOR CHAPTER 5 Absorbing Markov chain 404 Absorbing state 404 Characteristic polynomial of a linear operator 249 Characteristic polynomial of a ma-trix 218 Column sum of a matrix 295 Convergence of matrices 284 Cyclic subspace .'514 Diagonalizable linear operator 245 Diagonalizable matrix 246 Direct sum of matrices 320 Direct sum of subspaces 275 Eigenspace of a linear operator 264 Eigenspace of a matrix 264 Eigenvalue of a linear operator 246 Eigenvalue of a matrix 246 Eigenvector of a linear operator 246 Eigenvector of a matrix 246 Fixed probability vector 301 Generator of a cyclic subspace 313 Gerschgorin disk 296 Initial probability vector for a Markov chain 292 Invariant, subspace 313 Limit of a sequence of mat rices 284 Markov chain 291 Markov process 291 Multiplicity of an eigenvalue 263 Probability vector 289 Regular transition matrix 294 Row sum of a matrix 295 Splits 262 Stochastic process 288 Sum of subspaces 275 Transition matrix 288 6 I n n e r P r o d u c t S p a c e s 6.1 6.2 Inner Products and Norms The Gram-Schmidt Orthogonalization Process and Orthogonal Complements The Adjoint of a Linear Operator Normal and Self-Adjoint Operators Unitary and Orthogonal Operators and Their Matrices Orthogonal Projections and the Spectral Theorem The Singular Value Decomposition and the Pseudoinverse Bilinear and Quadratic Forms Einstein's Special Theory of Relativity 6.10* Conditioning and the Rayleigh Quotient 6.11* The Geometry of Orthogonal Operators 6.3 6.4 6.5 6.6 6.7* 6.8* 6.9* M , Lost applications of mathematics are involved with the concept, of mea-surement and hence of the magnitude or relative size of various quantities. So it is not surprising that the fields of real and complex numbers, which have a built-in notion of distance, should play a special role. Except for Section 6.8, we assume that all vector spaces are over the field F, where F denotes either R or C. (See Appendix D for properties of complex numbers.) We introduce the idea of distance or length into vector spaces via a much richer structure, the so-called inner product spaa structure. This added structure provides applications to geometry (Sections 6.5 and 6.11), physics (Section 6.9), conditioning in systems of linear equations (Section 6.10), least squares (Section 6.3), and quadratic forms (Section G.8). 6.1 INNER PRODUCTS AND NORMS Many geometric notions such as angle, length, and perpendicularity in R~ and R' may be extended to more general real and complex vector spaces. All of these ideas are related to the concept of inner product. Definition. Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors .r and y in V. a 329 330 Chap. 6 Inner Product Spaces scalar in F, denoted (x,y), such that for all x. y, and z in V and all c in F, the following hold: (a) (x + z,y) = (x,y) + (z,y). (b) (cx,y) = c(x,y). (c) (x,y) = (y,x), where the bar denotes complex conjugation. (d) (x,x) X ) ifxi= 0. Note that (e) reduces to (x,y) = (y,x) if F = R. Conditions (a) and (b) simply require that the; inner product be linear in the first component. It. is easily shown that if a .a2,... , an E F and y, v, v2,... , vn E V. t hen (^aiVi,y) = ^2at (vt.y). Example 1 For x = (ai, a2..... an) and y = (b, b2...... bn) in F'. define n fay) = ^2'i'>i-i i The verification that ( ) satisfies conditions (a) through (d) is easy. For example, if z (c. c2-. c-n). we have for (a.) (x 4- z, y) - ^2(a-i + Ci)k = ^ a. J), + ^ c,b, i=l i=l i=l = fa y) + (z,. y). Thus, for x = (1 4- i,4) and y = (2 - 3i,4 I 5i) in C2, (x, y) = (i + i)(2 4- 30 + 4(4 - 5i) = 1 5 - 1 5 1 The inner product in Example 1 is called the standard inner product on F'. When F R the conjugations are not needed, and in early courses this standard inner product is usually called the dot product and is denoted by X'-y instead of {x,y). Example 2 If (;/;, y) is any inner product on a. vector space; V and r > 0, we may define another inner product by the rule (x,y) = r{x.y). If r < 0, then (d) would not hold. Sec. 6.1 Inner Products and Norms 331 Example 3 Let V C([0,1]), the vector space of real-valued continuous functions on [0,1]. For f.g C V. define (f.g) - ][] f(t)g(t)dt. Since the preceding integral is linear in / . (a) and (b) are immediate, and (c) is trivial. I f / / tl. then f2 is bounded away from zero on some subinterval of [0,1] (continuity is used here), and hence (/. / ) = /J [f(t)]2 dt > 0. Definition. Let A E Mm X n (F) . We define the conjugate transpose or adjoint of A to be the n x m matrix A* such that (A')ij A J; for all i.j. Example 4 Let Then A = i 1 4-2/ 2 3 + U A* = 1 - 2/ 3 i Notice that if x and y are viewed as column vectors in F'. then (x,y) = //' The conjugate transpose of a matrix plays a very important role in the remainder of this chapter. In the case that .4 has real entries. A': is simply the I ranspose of A. Example 5 Let V - MX(F). and define (.4. B) - tv(B'A) for A.B E V. (Recall that the trace of a matrix A is defined by tr(.l) Y]'_ , An.) We verify that (a) and (d) of the definition of inner product, hold and leave (b) and (e) to the reader, for this purpose, let. A,B,C C V. Then (using Exercise (> of Section 1.4) (.4 + B. C) = tr(C*(/l 4- B)) - t r (CM 4- C*B) - triC* A) r- tr(C*B) = (A.C) + {B.C). Also (A. A) = tr(AM) = ^2(A*A)u = X ) E ( A * ) * * A i-- ; I.- i 2 i=] fc=l M , A , , - J : E ^ / - I &=] Now if .4 ^ (). then Aki ^ 0 for some k and i. So (A, A) > 0. 332 Chap. 6 Inner Product Spaces The inner product on M n X n (F) in Example 5 is called the Frobenius inner product. A vector space V over F endowed with a specific inner product is called an inner product space. If F C, we call V a complex inner product space, whereas if F = R. we call V a real inner product space. It is clear that if V has an inner product (x,y) and W is a subspace of V, then W is also an inner product space when the same function (.x, y) is restricted to the vectors x. y E W. Thus Examples 1, 3, and 5 also provide examples of inner product spaces. For the remainder of this cha,pter, Fn denotes the inner product space with the standard inner product as defined in Exam.ple 1. Likewise, Mnxn(F) denotes the inner product space with the Frobenius inner product as defined in Example 5. The reader is cautioned that two distinct inner products on a given vector space yield two distinct inner product spaces. For instance, it can be shown that both (f(x),g(x)), = 1 f(t)g(t)dt and (f(x),g(x))2 = j f(t)g(t)dt are inner products on the vector space P(R). Even though the underlying vector space is the same, however, these two inner products yield two different inner product spaces. For example, the polynomials f(x) = x and g(x) x2 are orthogonal in the second inner product space, but not in the first. A very important inner product space that resembles C([0, l|) is the space H of continuous complex-valued functions defined on the interval [0, 27t] with the inner product (f,g) = I 2TT f(t)g(t)dt. The reason for the constant 1/27T will become evident later. This inner prod-uct space, which arises often in the context of physical situations, is examined more closely in later sections. At this point, we mention a few facts about integration of complex-valued functions. First, the imaginary number /' can be treated as a constant under the integration sign. Second, every complex-valued function / may be written as f fi -- if2, where f and f2 are real-valued functions. Thus we have jf=]fi+ijf2 and / / = / / From these properties, as well as the assumption of continuity, it follows that H is an inner product space (see Exercise 16(a)). Some properties that follow easily from the definition of an inner product are contained in the next theorem. Sec. 6.1 Inner Products and Norms 333 Theorem 6.1. Let V be an inner product space. Then for x, y. 2 V and c E F. the following statements are true. (a) (x.y + z) = (x.y) faz). (b) facy) =c{x,y). (c) (x,0) = (0.x) = 0 . (d) (./../) = 0 if and only if x - 0. (e) //' {.r. //) = (./;. 2) for aii x E V. fcnen // z. Proof (a) We have {./-.// 4- ; ) - ( / / + z,x) = (y,x) + (z.x) = (y,x) + fax) = fay) + faz). The proofs of (b), (c), (d). and (e) are left as exercises. The reader should observe that (a) and (b) of Theorem (i.l show thai the inner product is conjugate linear in the second component. In order to generalize t he notion of length in R' to arbitrary inner product spaces, we need only observe that the length of x (a,b,c) C R' is given by s/a2 b- - c2 sj(x,x). This leads to the following definition. Definition. Let V be an inner product space. For x E V. we define the norm or length of x by x = /(x.x). Example 6 Let V F'. If ./ = (r/|. a-2 . . -a,,), then x : | |(ai,a2 . . . , a n ) | | --1 I /2 E 2 i^ the Euclidean definition of Length. Note that if // I. we have |||| lal. As we might expect, the well-known properties of Euclidean length in R'5 hold in general, as shown next. Theorem 6.2. Let V be an inner product space over F. Then for all x.y G V and c E F, the following statements are true. (a) ||cx|| = |c|-|k|l-(b) ||.r|| = 0 if and only if x = tl. In any ease, x > 0. (c) (Canchy Schwarz Inequality) {.iy) < ||:r|| y. (d) (Triangle Inequality) x y < ||.r|| 4- ||/y||. 334 Chap. 6 Inner Product Spaces Proof. We leave1 the proofs of (a) and (b) a.s exercises. (c) ]y = 0, then the result is immediate. So assume that y ^ 0. For any c E F , we have 0 < x cy (x cy, x cy) = (x, x - cy) c (y. x cy) = (x, x) - c (x, y) - c (y, x) f cc (y. y). In particular, if we set the inequality becomes fay) (Sec. 6.1 Inner Products and Norms 335 The reader may recall from earlier courses that, for x and y in R3 or R2, we have that (x.y) ||x||*|MI cos0, where 0 (0 < 6 < TT) demotes the angle between x and y. This equation implies (c) immediately since |cos#| < 1. Notice also that nonzero vectors x and y are perpendicular if and only if cos# = 0. that is. if and only if (x.y) = 0. We are now at the point where we can generalize the notion of perpendic-ularity to arbitrary inner product spaces. Definitions. Let V he an inner product space. Vectors x and y in V are orthogonal (perpendicular) if (x.y) = 0. A subset S ofV is orthogonal if any two distinct vectors in S are orthogonal. A vector x in V is a. unit vector ifx = 1. Finally, a subset S of V is orthonormal ifS is orthogonal and consists entirely of unit vectors. Note that if S = {v, v2....}, then S is orthonormal if and only if (vi,Vj) = Sij, where Sij denotes the Kronecker delta. Also, observe that multiplying vectors by nonzero scalars does not affect, their orthogonality and that if x is any nonzero vector, then (l/||x||)rr is a unit vector. The process of multiplying a nonzero vector by the reciprocal of its length is called normalizing. Example 8 In F3, {(1.1. 0), (1, 1, 1). (1,1, 2)} is an orthogonal set of nonzero vectors, but it is not orthonormal; however, if we normalize the vectors in the set, we obtain the orthonormal set { - L ( u , o ) , - ^ ( 1 , - 1 , 1 ) , 4 = ( - i , i , 2 ) } Our next example is of an infinite orthonormal set that is important in analysis. This set is used in later examples in this chapter. Example 9 Recall the inner product space H (defined on page 332). We introduce an im-portant orthonormal subset S of H. For what follows, i is the imaginary num-ber such that i2 = 1. For any integer n, let fn(t) eint, where 0 < t < 2-K. (Recall that eint count + isinnt.) Now define S = {/.: n is an integer}. Clearly S is a subset of H. Using the property that elt e ' for every real number t. we have, for m ^ n, /TOJ Jn) 0 2TT eimteint dt = 2TT 1 _'77 2iri(rn __ i(rn n)t ei(m-n)t dt = 0. 336 Also, Chap. 6 Inner Product Spaces In other words. {/,./.) = Smn. EXERCISES 1. Label the following statements as true or false. (a) An inner product is a scalar-valued function on the set of ordered pairs of vectors. (b) An inner product space must, be over the field of real or complex numbers. (c) An inner product is linear in both components. (d) There is exactly one inner product on the vector space R'. (e) The triangle inequality only holds in finite-dimensional inner prod-uct, spaces. (f) Only square matrices have a conjugate-transpose. (g) If x. y. and z are vectors in an inner product space such that (x,y) (x.z), then y z. (h) If (x,y) 0 for all x in an inner product space, then y = 0. 2. Let. x = (2.1 f i,i) and y - (2 v',2. 1 4-2/) be vectors in C3. Compute (x.y). x, y, and ||:r + v/||. Then verify both the Cauchy Schwarz inequality and the triangle inequality. 3. In C([0, 1]), let f(t) - I and g(t) = e'. Compute (fg) (as defined in Example 3). |j/||, '|Sec. 6. 7. 9. 10. 11. 12. 6.1 Inner Products and Norms 337 Complete the proof of Theorem 6.1. Complete the proof of Theorem Ci.2. Provide reasons why each of the following is not an inner product on the given vector spaces. (a) ((a.b).(c.d)) =ac bd on R2. (b) (A, B) = tv(A 4- B) on U2x2(R). (c) (f(x),g(x)) = / J f'(t)g(t)dt on P(R), where ' denotes differentia-tion. Let 0 be a basis for a finite-dimensional inner product space. (a) Prove that if (x, z) = 0 for all z E ft. then x = 0. (b) Prove that if (x. z) = (y, z) for all z E 0, then x = y. Let V be an inner product space, and suppose that X and y are orthog-onal vectors in V. Prove that x 4- y2 = ||a:||2 4- y2 Deduce the Pythagorean theorem in R2. Prove the parallelogram law on an inner product space V: that is, show that x + y2 + x - y2 = 2||;c||2 4- 2y2 for all x. y E V. What does this equation state about parallelograms in R2? Let {v. v2,..., 338 Chap. 6 Inner Product Spaces and let z = x ay. Prove that y and z are orthogonal and a = //I Then apply Exercise 10 to ||.r||2 = ay 4- z2 to obtain ||c|| = 0. (b) Derive a similar result for the equality ||x 4- y = ||.r|| 4- y- and generalize; it to the case of /; vectors. 16. (a) Show that the vector space H with (, ) defined on page 332 is an inner product space, (b) Let V - C([(),l]). and define ,1/2 ( .A//)- / f(t)g(t)dt. Is this an inner product on V? 17. Let T be a linear operator on an inner product space V. and suppose that ||T(;?r)|| = ||;/:|| for all x. Prove that T is one-to-one. 18. Let V be a vector space over /'. where F = R or F - ('. and let W be an inner product space ewer F with inner product (. ). If T: V > W is linear, prove that (x.y) (T(x).T(y)) defines an inner product on V if and only if T is one-to-one. 19. Let V be an inner product space. Prove that (a) x y2 = x2 24? fa y) + y2 for all x. y E V, where 3? (x. y) denotes the real part of the complex number (x.y). (t>) I ikll W i < ll;r jy'| for all x,y G V. 20. Let V be an inner product space over F. Prove the polar identities: For all x, yE V, (a) fay) = lx + y2-x- y2 if F = R; (b) fay) = E L I Sec. 6.1 Inner Products and Norms 339 22. Let V be a real or complex vector space (possibly infinite-dimensional), and let 3 be a basis for V. For x.y E V there exist v,v2.... ,v E ft such that Define x = 2_]aivi a n ( i y = /^hVj-I I i= fay) = ])2'i.bi. (a) Prove that ( ) is an inner product, on V and that 0 is ail or-thonormal basis for V. Thus every real or complex vector space; may be regarded as an inner product space. (b) Prove that if V R' or V = C' and ft is the standard ordered basis, then the inner product, defined above is the standard inner product. 23. Let V = Fn, and let A E M.X/I(/'). (a) Prove that (x. Ay) = (A*x,y) for all x,y C V. (b) Suppose that for some B M n x n (F ) , we have (x,Ay) = (Bx,y) for all x.y E_ V. Prove that B = A*. (c) Let a be the standard ordered basis for V. For any orthonormal basis 0 for V, let Q be the n x n matrix whose columns are the vectors in ft. Prove that. Q* =Q ' . (d) Define linear operators T and U on V by T(x) Ax and U(x) = A*x. Show that [J]p [T]g for any orthonormal basis ft for V. The following definition is used in Exercises 24 27. Definition. Let V be a vector space over F, where F is either R or C. Regardless of whether V is or is not an inner product space, we may still define a norm. || as a. real-valued function on V satisfying the following three conditions for all x. y C V and a E F: (1) j|.r|| > 0. and x = 0 if and only if x = 0. (2) ax = a ||.r||. (3) x + y < x + y. 24. Prove that the following are norms on the given vector spaces V. (a) V - M m x n (F) ; ||^|| = max AKj for all ,4 G V (b) V - C([0,11); f = max 1/(01 for all / G V t[0,l] 340 Chap. 6 Inner Product Spaces (c) V = C([0,1]); 11/11=/ f(t)dt f o r a l l / G V (d) V = R2; i. b) = max{|a|, j6|} for all (a,b) E V 25. Use Exercise 20 to show that there is no inner product (, ) on R2 such that x2 = (x,x) for all x E R2 if the norm is defined as in Exercise 24(d). 26. Let || |j be a norm on a vector space V, and define, for each ordered pair of vectors, the scalar d,(x.y) = x y, called the distance between x and y. Prove the following results for all x, y, z E V. (a) d(a; ,y)>0. (b) d(x,y) = d(y,x). (c) dfay)< dfaz) + dfay). (d) dfa x) = 0. (c) dfa y) f 0 if x ^ y. 27. Let || || be a. norm on a real vector space V satisfying the parallelogram law given in Exercise 11. Define fay) = 4 [lls + 3/ll2- lk-2/ l l2 ] Prove that (, ) defines an inner product on V such that ||.x||2 = (x,x) for all x E V. Hints: (a) Prove (x, 2y) = 2 (x, y) for all x. y E V. (b) Prove (x 4- u, y) (x, y) + (u. y) for all x, a. y E V. (c) Prove (nx.y) = ?i(x,y) for every positive integer n and every x,y E V. (d) Prove m(x,y) = (x,y) for every positive integer m and every x.yEV. (e) Prove (rx, y) = r (x, y) for every rational number r and every x, y E V. (f) Prove | (x.y) | < ||.'c||||y|| for every x.y E V. Hint: Condition (3) in the definition of norm can be helpful. (g) Prove that for every c E R, every rational number r, and every x,yV, cfay) - (cx,y) | = | ( c - r ) fay) - ((cr)x,y) < 2|c-r|||:e||||y||. (h) Use the fact that for any c E R. c r can be made arbitrarily small, where r varies over the set of rational numbers, to establish item (b) of the definition of inner product. Sec. 6.2 Gram-Schmidt Orthogonalization Process 341 28. Let V be a complex inner product space with an inner product (, ). Let [, ] be the real-valued function such that [x.y] is the real part of the complex number (x.y) for all x.y E V. Prove that [*, ] is an inner product for V, where V is regarded as a vector space over R. Prove, furthermore, that [x,ix] = 0 for all x E V. 29. Let V be a vector space over C, and suppose that [, ] is a real inner product on V. where V is regarded as a vector space over R. such that [.;.-,/.;] = 0 for all ./ G V. Let ( ,) be the complex-valued function defined by Prove that fa y) = [./:, y] 4- i[x, iy] for x. y c V. ) is a complex inner product on V. 30. Let || || be a norm (as defined in Exercise 24) on a complex vector space V satisfying the parallelogram law given in Exercise 11. Prove that there is an inner product (, ) on V such that ||aj|| = (x,x) for all x G V. Hint: Apply Exercise 27 to V regarded as a vector space over R. Then apply Exercise 29. 6.2 THE GRAM-SCHMIDT ORTHOGONALIZATION PROCESS AND ORTHOGONAL COMPLEMENTS In previous chapters, we have seen the special role of the standard ordered bases for C' and R'. The special properties of these bases stem from the fact that the basis vectors form an orthonormal set. .lust as bases are the building blocks of vector spaces, bases that are also orthonormal sets are the building blocks of inner product spaces. We now name such bases. Definition. Let V be an inner product space. A subset of V is an orthonormal basis forW if it is an ordered basis that is orthonormal. Example 1 The standard ordered basis for F' is an orthonormal basis for F'. Example 2 The set is an orthonormal basis for R2. 342 Chap. 6 Inner Product Spaces The next theorem and its corollaries illustrate why orthonormal sets and. in particular, orthonormal bases are so important. Theorem 6.3. Let V be an inner product, space and S {v, v2 ,vk] be an orthogonal subset of/ consisting of nonzero vectors. If y E span(S), then y,Vi i=i '/ Proof. Write y 2_, a>vi.i where a,, a2,..., ak G F. Then, for I < j < k. i^i we have (y,vj) = Y^(HVi-v>) = lLai^'^VJ) = ai (vJ>vl) = ajvj2-So aj .32 , and the result follows. I The next corollary follows immediately from Theorem G.3. Corollary 1. If, in addition to the hypotheses of Theorem 6.3, S is orthonormal and y E span(S), then k y = ^2(y>Vi)vi. i= If V possesses a finite orthonormal basis, then Corollary 1 allows us to compute the coefficients in a linear combination very easily. (See Example 3.) Corollary 2. Let V be an inner product space, and let S be an orthogonal subset of'V consisting of nonzero vectors. Then S is linearly independent. Proof. Suppose that V, t^2-, ,nk E S and y^aiVt = o. i=l As in the proof of Theorem 6.3 with y ~ 0. we have o.j = (0, Vj) for all j. So S is linearly independent. Ie.,ll2=0 Sec. 6.2 Gram-Schmidt Orthogonalization Process 343 Example 3 By Corollary 2. the orthonormal set ^ (1 ,1 ,0) i ( 1 , -1 ,1) , 4 g ( - l , l , 2 ) obtained in Example 8 of Section 6.1 is an orthonormal basis for R'!. Let x = (2.1.3). Fhe coefficients given by Corollary 1 to Theorem 0.3 that express x as a linear combination of the basis vectors an; a i = - L ( 2 + l) = - L a 2 = J - ( 2 _ 1 + 3 ) = - l and As a check, we have 3 = - 7 = ( - 2 + 1 4-6) - =. /6 v6 (2,1,3) = | (1 ,1 ,0) + | ( 1 , -1 ,1 ) + | ( - 1 , 1 , 2 ) . Corollary 2 tells us that the vector space H in Section 6.1 contains an infinite linearly independent set, and hence H is not a finite-dimensional vector space. Of course, we have not. yet shown that, every finite-dimensional inner prod-uct space possesses an orthonormal basis. Fhe next theorem takes us most of the way in obtaining this result. It tells us how to construct an orthogonal set from a linearly independent, set of vectors in such a way that both sets generate the same subspace. Before stating this theorem, let us consider a simple case. Suppose that {W,W2) is a linearly independent, subset of an inner product space (and hence a basis for some two-dimensional subspace). We want to construct an orthogonal set from {w.w344 Chap. 6 Inner Product Spaces W = Vl Figure 6.1 The next theorem shows us that this process can be extended to any finite linearly independent subset. Theorem 6.4. Let V be an inner product space and S = {w. w2 , Wn } be a linearly independent, subset o/'V. Define S' = {i',v2. .. ,vn}. where Vi = W and AI for 2 < k < n. (1) '3 Then S' is an orthogonal set of nonzero vectors such that span(S') = span(5). Proof. The proof is by mathematical induction on n, the number of vectors in 5. For k 1,2, . . . , n , let Sk {w.w2 ,wk}. If n = 1. then the theorem is proved by taking S[ = S|; i.e., Vi = Sec. 6.2 Gram-Schmidt Orthogonalization Process 345 Example 4 In R'. let wj --- (1,0,1,0), w2 - (1,1,1,1), and w3 = (0,1,2,1). Then [wi,w2,w3} is linearly independent. We use the Cram Schmidt process to compute the orthogonal vectors 346 Furthermore, Therefore Chap. 6 Inner Product Spaces = / t2-ldt=- and (x2,v2) = /2 -tdt = 0. ./-i V3 X Sec. 6.2 Gram-Schmidt Orthogonalization Process 347 Proof. Let 0o be an ordered basis for V. Apply Theorem 6.4 to obtain an orthogonal set 0' of nonzero vectors with span(,#') = span(/?n) = V. By normalizing each vector in 0', we obtain an orthonormal set ft that generates V. By Corollary 2 to Theorem 6.3, ft is linearly independent; therefore ft is an orthonormal basis for V. The remainder of the theorem follows from Corollary I to Theorem 6.3. Example 6 We use Theorem 6.5 to represent the polynomial f(x) = 14- 2x 4- Sx as a linear combination of the vectors in the orthonormal basis {1*1,1*2,1*3} for P2{T{) obtained in Example 5. Observe that /i i ( /( . ,-) .,)- / - = ( l + 2/. + 3c2)c/i-2v /2, .1 - v2 2 ^ = t(i + 2t + 'M2)dt-and ( / ( * W = -(3/' 1 2t + SP)dt = 2v/10 r 2Vo 2vT0 Therefore f(x) 2J2 u - u2 H 1*3. 3 0 Theorem 6.5 gives us a simple method for computing the entries of the matrix representation of a linear operator with respect to an orthonormal basis. Corollary. Let V be a finite-dimensional inner product space with an orthonormal basis 0 = {('1, v2,.... vn}. Let T be a linear operator on V, and let A [T]p. Then for any i and j, Aj.j (T(vj).Vi). Proof From Theorem 6.5, we have; Hence Aij = {T(vj),vi). I The scalars ('.//) given in Theorem 6.5 have been studied extensively for special inner product spaces. Although the vectors V ,v2, ,vn were chosen from an orthonormal basis, we introduce a terminology associated with orthonormal sets 0 in more1 general inner product spaces. 348 Chap. 6 Inner Product Spaces Definition. Let ft be an orthonormal subset (possibly infinite) of an inner product space V, and let x E V. We define the Fourier coefficients of x relative to ft to be the scalars (x.y). where y E ft. In the first half of the 19th century, the French mathematician Jean Bap-tiste Fourier was associated with the study of the scalars 2ir f(t) sin ntdt and f(t)cosntdt, ./o or more generally. 2n 2TT /(*)' rint dt. for a function / . In the context of Example 9 of Section 6.1, we see that cn = (fifn), where fn(t) e'1: that is. cn is the nth Fourier coefficient for a continuous function / G V relative to S. These coefficients are the 'classical' Fourier coefficients of a function, and the literature concerning the; behavior of these coefficients is extensive. We learn more about these1 Fourier coefficients in the remainder of this chapter. Example 7 Let S = {eint: n is an integer}. In Example 9 of Section 6.1, S was shown to be an orthonormal set, in H. We compute the Fourier coefficients of f(t) = t relative to S. Using integration by parts, we have, for n ^ 0, (f,fn) = -e 2 71 *2TT tc'> dt = ./ll 1 2^ 'ii dt = - 1 171 and. for n = 0. I E 71= fc -1 E n = -k k if-In) Z-^t T)2 ( / i ) r + x ; i { / - / , } i 2 fc y -n=l n= Sec. 62 Gram-Schmidt Orthogonalization Process for every k. Now. using the' fact that |!/||2 = it , we obtain 349 U 2 > 2 Y + 7T2, rr- I 6 > E -Z_^ ,,2 Because this inequality holds for all A:, we may let k > oo to obtain 2 oo x 6 - Z ^ ,2-Additional results may be produced by replacing / by other functions. We' are' now ready to proceed with the; concept of an orthogonal comple-ment. Definition. Let S be a nonempty subset of an inner product space V. We define S (read 'S perp') to be the set of all vectors in V that are orthogonal to every vector in S: that, is, S' {x E V: (x, y) = 0 for all y E S}. The set S is called the orthogonal complement of S. It is easily seen that S ' is a subspace' of V for any subset S of V. Example 8 Fhe reader should verify that {()}~L = V and V ' {0} for any inner product space V. Example 9 If V = R'5 and S {e3}, then S1- equals the xi/-plane (see Exercise 5). Exercise 18 provides an interesting example of an orthogonal complement in an infinite-dimensional inner product, space. Consider the problem in R,J of finding the distance from a point P to a plane W. (See figure 6.2.) Problems of this type arise in many settings. If we let y be the vector determined by 0 and P. we may restate the problem as follows: Determine the; vector a in W that is 'closest' to y. The desired distance is clearly given by y u. Notice from the figure that the vector z = y a is orthogonal to every vector in W, and se> z G W 1 . The next result presents a practical method of finding u in the case that W is a finite-dimensional subspace of an inner product, space. ' 350 Chap. 6 Inner Product Spaces = y - ' w Figure 6.2 Theo rem 6.6. Let W be a finite-dimensional subspace of an inner product space V, and let y E V. Then there exist unique vectors u E W and z E W1 such that y = u + z. Furthermore, if {v ,v2. vk} is an orthonormal basis for W, then k u = y^(y,Vi)vj. i=l Proof. Let {vi,v2,...,vk} be an orthonormal basis for W. let u lie as defined in the preceding equation, and let z = y a. Clearly a E W and y = u + z. To show that z E W-1-, it suffices to shew, by Exercise 7, that z is orthog-onal to each Vj. For any j, we have fc [Z,Vi) = y- 2_s(y>vi)vi >VJ = (V,VJ) - (y,Vj) = 0 . [y>vi) - J2(y>vi)(vi,vj) i i To show uniqueness of u and z, suppose that y = u + z = a' + z'. where u' E W and z' E W . Then a - v! = z' - z E W n W 1 - {()}. Therefore, u = u' and z = z'. I Corollary. In the notation of Theorem 0.6. the vector u is the unique vector in W that is 'closest' to y; that is, for any x E W. y - x > y - u, and this inequality is an equality if and only if x = a. Proof. As in Theorem 6.6, we have that y = u+ z, where z E NL. Let x E W. Then t* x is orthogonal to z, so, by Exercise 10 of Section (i.l. we-Sec. 6.2 Gram-Schmidt Orthogonalization Process have 351 y x2 u + z x2 > ||2||2 = y- u2. [u - x) + z2 = u - x2 + z2 Now suppose that y - x = y u. Then the inequality above becomes an equality, and therefore u #||2 + ||z||2 = ||z||2. It follows that ||w x = 0, and hence x = u. The proof of the converse is obvious. The vector u in the corollary is called the orthogonal projection of y on W. We will see the importance; of orthogonal projections of vectors in the application to least squares in Section 6.3. Example 10 Let V = P;i(R) with the inner product (f(x).g(x)) - j f(t)g(t)dt for all f(x),g(x) G V. We compute the orthogonal projection f(x) of f(x) = x3 on P2(R)-By Example 5. {ui,U2, U3} ^ V 2 V 8 v is an orthonormal basis for P2(^?)- For these vectors, we have , 1 (f(x),Ul)=J^t3-j=dt = 0, = , 3 , V6 and Hence ~ I tl-(3t'-l)dt = 0. fx{x) = (f(x), U]) u, + (f(x), u2) u2 + (f(x),u3) u3 = -x. It was shown (Corollary 2 to the replacement theorem, p. 17) that any lin-early independent set in a finite-dimensional vector space can be extended to a basis. The next theorem provides an interesting analog for an orthonormal subset of a finite-dimensional inner product space. 352 Chap. 6 Inner Product Spaces IT Theorem 6.7. Suppose that S = {i>. v-2- ..,?'/,} is an orthonormal set in an n-diinensional inner product space V. Then (a) S can be extended to an orthonormal basis {v, v2,.... Vk Vk+1, - , vn} for V. (b) If W = span(S'), then Si = {Vk+i,Vk+2, ivn} is an orthonormal basis for W ' (using the preceding notation). (c) If W is any subspace ofV, then dim(V) = dini(W) + diin(WL). Proof, (a) By Corollary 2 to the replacement theorem (p. 47), S can be extended to an ordered basis S' = [v, v2, Vk, Wk -i, , % } for V. Now apply the Gram Schmidt process to S'. The first k vectors resulting from this process are the vectors in .S1 by Exercise 8, and this new set spans V. Normalizing the last n k vectors of this set. produces an orthonormal set that spans V. The result now follows. (b) Because S is a subset of a basis, it is linearly independent. Since S is clearly a subset of W- . we need only show that it spans W 1 . Note that, for any x 6 V. we have x = 2^{X,Vi)Vi. lix e W 1 , then (x, v{) - 0 for 1 < i < k. Therefore, n x = ^2 (x, Vj) Vi span(5]). i=k+l (c) Let W be a subspace of V. It is a finite-dimensional inner product space because V is, and so it has an orthonormal basis {v,V2, :t'fc}- By (a) and (b). we have dim(V) - n = k + (n - k) = dim(W) + dimfVV1). | Example 11 Let W = span({ei,e2}) in F3. Then x - {a.b.c) e W 1 if and only if 0 = (ar,ei) = a and 0 = (x,e2) = b. So x = (0,0,c), and therefore W- = span({c,{}). One can deduce the same result by noting that e3 W'1 and, from (c), that din^W-1) = 3 - 2 = 1. EXERCISES 1. babel the following statements as true or false. (a) The Gram Schmidt orthogonalization process allows us to con-struct an orthonormal set from an arbitrary set of vectors. Sec. 6.2 Gram-Schmidt Orthogonalization Process 353 (b) Every nonzero finite-dimensional inner product space has an or-thonormal basis. (c) The orthogonal complement of any set is a subspace. (d) If {(,], 1*2,.... vn} is a basis for an inner product space V, then for any x V the scalars (x, c,) are the Fourier coefficients of x. (e) An orthonormal basis must be an ordered basis. (f) Every orthogonal set is linearly independent. (g) Every orthonormal set is linearly independent. 2. In each pari, apply the Gram Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S'). Then normalize the vectors in this basis to obtain an orthonormal basis ri for span(S'). and compute the Fourier coefficients of the given vector relative to 0. Finally, use Theorem 6.5 to verify your result. (a) V - R: S - {(1.0, 1).(0. I. I). (1.3.3)}, and x = (1,1,2) (b) V - R;!. S = {(1, 1, I), (0,1, I), (0,0,1)}, and x - (1,0,1) (c) V - P2(/?) with the inner product (f(x),g(x)) - J* f(t)g(t)dt, S= {..r..r2}. and h(x) = 1+X (d) V - span(.S'). where S = {(1,2,0), (1 -1,2,4*)}, and x = (3 + i,4i, 1) (e) V - R1. S - {(2. 1,-2,4), ( - 2 ,1 , -5 ,5 ) , (-1,3,7,11)}, and x = ( -11.8 . -4 .18) (f) V = R4, S= {(1,-2, -1 ,3) , (3,6,3, 1), (1,4,2,8)}, and x = ( -1,2,1,1) / , K, / i ,354 Chap. 6 Inner Product Spaces (1) V = M2X2(C), 5 = - 2 5 - 3 8 / - 2 - 13/ 12 -78 / -7 - r24 / > i ) V - M2x2(C). 5 = 1 /' - 2 - 3/ / 8i 2 I 2/ 1 + /' J ' 1 - 3 - 3 / and A - I I i 2 i 11 - 132/ - 3 4 - 3 1 / 7 126/ -71 - 5/ I | 3/ . and A - 2 + 8/ - 1 3 i 10-10/ 9 - 0 / - 1 - 7 / - 9 - 8i' 1 -r 10?: - 6 - 2 / ' ; - 7 I 5i 3 + 18/ 9 - 6/ - 3 + 7/ I - 1 9' /0 3. In R2, lei ' - ( ( 4 = . 4= Find (he Fourier coefficients of (3,4) relative to 0 4. Let S = {(1,0,i), (1,2,1)} in C3. Compute S1. 5. Let S ./u}- where XQ is a nonzero vector in R,{. Describe Sn ge-ometrically. Now suppose that S {x%,x2} is a linearly independent subset of R3. Describe S geometrically. 6. Let V be an inner product space, and let W be a finite-dimensional subspace of V. If .; (f: W, prove that there exisls y c V such that y e WJ . but (.r.y) / 0. Hint: Use Theorem 0.6. 7. Let 0 be a basis lor a subspace W of an inner product space V. and let z e V. Prove thai : f W : if and only if (z, v) = 0 for every /' G 3. 8. Prove that if { a-. w2 ir } is an orthogonal set of nonzero vectors. then the vectors V, r2 /' derived from the (Irani Schmidt process satisfy v, = ir, for i = 1.2 u. Hint.: Use mathematical induction. 9. Let W - span({(/'.(). 1)}) in C:{. Find orthonormal bases for W and W^. 10. Let W be a. finite-dimensional subspace of an inner product space V. Prove that there exists a projection T on W along W thai satisfies ||(T) = W 1 . In addition, prove that ||T(J-)|| < ll-'| for all .; e V. Hint: Use Theorem 6.6 and Exercise 10 of Section 0.1. (Projections are defined in the exercises of Sect ion 2.1.) 11. Let A be an // x /; matrix with complex entries. Prove thai .LP = / if and only if the rows of A form an orthonormal basis for C'. 12. Prove that for any matrix A ; n ( R ( U . ) r =N(L, Sec. 6.2 Gram-Schmidt Ort nalization Process 355 13. Let V be an inner product space, S and SQ be subsets of V. and W be a finite-dimensional subspace of V. Prove the following results. (a) So C 5 implies that 5 X C S(f. (b) S C (6 , ) x ; so span(S) C ( 5 ) . (c) W = (W ' J1 , fltnt; Use Exercise 6. (d) V W W^. (See the exercises of Section 1.3.) 14. Let Wi and W2 be subspaces of a. finite-dimensional inner product space. Prove that (W, + W2) ' = W f n W ^ and (W, nW 2 ) L = W,1 + W2L. (See the definition of the sum of subsets of a vector space on page 22.) Hint for the second equation: Apply Exercise 13(c) to the first equation. 15. Let V be a finite-dimensional inner product space over F. (a) Parseval's Identity. Let {u ,v2,... ,vn} be an orthonormal basis for V. For any x. y G V prove that n (x.y) ='^2{x,Vi)(y,Vi}. i-l (b) Use (a) to prove that if 0 is an orthonormal basis for V with inner product (, ), then for any x.y 6 V (Mx),My)Y = ([x}0,[ybY = (x,y), *' where {. ) is the standard inner product, on F'. 16. (a) Bessel's Inequality. Let V be an inner product space, and let S = {r,. c 2 . . . . . vn } be an orthonormal subset of V. Prove that for any x V we have Ml2 > l !' Hint: Apply Theorem 6.6 to x e V and W = span(5). Then use Exercise 10 of Section 6.1. (b) In the context of (a), prove that Bessel's inequality is an equality if and only if x G span(5). 17. Let T be a linear operator on an inner product space V. If (T(x),y) = 0 for all .r.y G V, prove that T = To. In fact, prove this result if the equality holds for all x and y in some basis for V. 18. Let V = C([- l , 1]). Suppose that W,. and W denote the subspaces of V consisting of the even and odd functions, respectively. (See Exercise 22 356 Chap. 6 Inner Product Spaces of Section 1.3.) Prove that Wg W0, where the inner product on V is defined by > = / MMVdt. 19. In each of the following parts, find the orthogonal projection of the given vector on the given subspace W of the inner product space V. (a) V = R2, u = (2,6), and W = {(x, y): y = Ax}. (b) V = R3, u = (2,1, 3), and W = {{x. y, z): x + 3y - 2z = 0}. (c) V = P(R) with the inner product. (f(x),g(x)) = J0' / (%(*) * , /i(.7;) = 4 + 3.7; - 2a;2, and W = P, (R). 20. In each part of Exercise 19, find the distance from the given vector to the subspace W. 21. Let V = C([- l , 1]) with the inner product (f,g) = f f(t)g(t)dt, and let W be the subspace P2(i?), viewed as a space of functions. Use the orthonormal basis obtained in Example 5 to compute the 'best' (closest) second-degree polynomial approximation of the function h(t) = el on the interval [1,1]. ..' 22. Let V = C([0,1]) with the inner product. (f,g) = / J f(t)g(t) dt. Let W be the subspace spanned by the linearly independent set {/, /t}. (a) Find an orthonormal basis for W. (b) Let h(t) = t2. Use the orthonormal basis obtained in (a) to obtain the 'best' (closest) approximation of //. in W. 23. Let V be the vector space defined in Example 5 of Section 1.2, the space of all sequences a in F (where F = R or F = C) such that o~(n) 7^ 0 for only finitely many positive integers n. For rr, fi G V. we define (a,fi) = y o-(n)fi(n). Since all but a finite number of terms of n=] the series are zero, the series converges. (a) Prove that (, ) is an inner product on V, and hence V is an inner product space. (b) For each positive integer n, let en be the sequence defined by f-n(k) = k is the Kronecker delta. Prove that {ci, e2,...} is an orthonormal basis for V. (c) Let an = e + en and W = span({ 2}. (i) Prove that eA $ W, so W V. (ii) Prove that W 1 = {()}, and conclude that W ^ (N1). Sec. 6.3 The Adjoint of a Linear Operator 357 Thus the assumption in Exercise 13(c) that W is finite-dimensional is essential. 6.3 THE ADJOINT OF A LINEAR OPERATOR In Section 6.1. we defined the conjugate transpose A* of a matrix A. For a linear operator T on an inner product space V. we now define a related linear operator on V called the adjoint of T. whose matrix representation with respect to any orthonormal basis 0 for V is [T]*?. The analogy between conjugation of complex numbers and adjoints of linear operators will become apparent. We first need a preliminary result. Let V be an inner product space, and let y G V. The function g: V F defined by g(.r) = (x,y) is clearly linear. More interesting is the fact that if V is finite-dimensional, every linear transformation from V into F is of this form. Theorem 6.8. Let V be a finite-dimensional inner product space over F, and let g: V * F be a linear transformation. Then there exists a unique vector y G V such that g(x) = (x. y) for all x G V. Proof. Let 0 = {i?i,v2 , . . . . vn) be an orthonormal basis for V, and let. r -i=l Define h: V > F by h(x) = (x.y), which is clearly linear. Furthermore, for 1 < j < w we have Hvj) = (vj,y) = (vj^gMvi) = Y^g(vi) (vj,vi) n ^gMSji =g(vj). i=l / ?'=! Since g and h both agree on 0, we have that g h by the corollary to Theorem 2.6 (p. 73). To show that y is unique, suppose that g(x) = (x, y') for all ;r. Then (x.y) = (x,y') for all x; so by Theorem 6.1(e) (p. 333). we have y = y'. Example 1 Define g: R2 > R by g(ai, a2) = 2ai +a 2 ; clearly g is a linear transformation. Let 0 = {ei,e2}, and let y = g(ex)e + g(e2)e2 = 2ex + e2 = (2,1), as in the proof of Theorem 6.8. Then g((i., a2) = {(ai, a2), (2, 1)) = 2a, + a2. 358 Chap. 6 Inner Product Spaces Theorem 6.9. Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Then there exists a unique function T*: V V such that (T(.r).u) = (./'. T'(y)) for all .r. y C V. furthermore. T' is linear. Proof. Lei // G V. Define g: V F by g(x) - (T(a:),y) for all s G V. We first show that g is linear. Let X,x2 V and c G F. Then g(cr, + x2) = (T(r.n + x2), y) = (cT(.r,) + T(.r2). y) - c(T(xl),y) i (T(.r2),//) - cg(x1) + g(.r2). Hence g is linear. We now apply Theorem 6.8 to obtain a unique vector ;/ V such that g(x) = (x.y'): that is. (T(.r).y) = {././/) for all x G V. Defining T* : V V by T'(y) - u'. we have (T(x).y) = (x,T'{y)). To show that T* is linear, let y,y2 C V and e G /'. Then for any x G V, we have (./.T* (r//i +r/2)) = (T(x),q/] +//_.) = r(T(.r),/y1) + (T(./-).//2) = e(*,T*(y1)) + (x,T*(ito)> = Sec. 6.3 The Adjoint of a Linear Operator 359 about adjoints. nevertheless, do not depend on V being finite-dimensional. Thus, unless stated otherwise, for the remainder of this chapter we adopt the convention that a reference to the adjoin! of a linear operator on an infinite-dimt nsional inner product space, assumes its existence. Theorem 6.10 is a useful result for computing adjoints. Theorem 6.10. Let V be a finite-dimensional inner product space, and let 3 be an orthonormal basis for V. IfT is a linear operator on V. then T 0 = V Proof. Let A = [T]p, D = [ T ' V and 0 = {v, v2. vn}. Then from the corollary to Theorem 6.5 (p. 346), we have B,j ~ (r(vj),Vi) = (i,T*(Vi)) = 360 Chap. 6 Inner Product Spaces (a) ( T + U ) * - T * + U*; (b) (cT)* = e.T* for any c G F; (c) (TU)* = U*T*: (d) T** = T: (e) P = I. Proof. We prove (a) and (d); the rest are proved similarly. Let x.y G V. (a) Because (x,(T + U)*(y)> - ((T4 U)(x),y) = Sec. 6.3 The Adjoint of a Linear Operator 361 feels that there exists an essentially linear relationship between y and t, say y ct + d, and would like to find the constants c and d so that the line y ct + d represents the best possible fit to the data collected. One such estimate of fit is to calculate the error E that represents the sum of the squares of the vertical distances from the points to the line; that is, E = X > - 362 Chap. 6 Inner Product Spaces first, we need some notation and two simple lemmas. For ./.// G F'. let (x,y) denote the standard inner product of .r and y in F'. Recall that if x and y are regarded as column vectors, then (x.y) = ;/'./'. Lemma 1. Let A < M,,(/') x c F'. and y t Fm. Then (Ax,y)m = (x,A*y)n. Proof. By a generalization of the corollary to Theorem (i.ll (see Exer-cise 5(1))). we have (Ax,y), //(,!./) (//,!)./ (A'y)mx= (x,A*y . I Lemma 2. Let A E M m x n (F ) . Then rank(^4*.4) rnnk(.-l). Proof. By the dimension theorem, we need only show that, for ./ G F'. we have A*Ax = 0 if and only if .b- tl. Clearly, Ax II implies that A*Ax=0. So assume that A*Ax t). Then 0 = {A*Ax,x)n = (Ax,A**x)m Ur.Ax)m. so that Ax = 0. I Corollary. If A is an in x n matrix such that iank(.l) n. then A' A is invertible. Xow let A he an in x // matrix and y F'. Define W {Ax: x C F'}: that is. W = R(L,i). By the corollary to Theorem 6.6 (p. 350). there exists a unique vector in W that is closest to y. Call this vector Axu. where ./,, r F'. Then ||Ax'o y < Ax y for all ./ G F'; so .; has the property that E Ax{) y is minimal, as desired. To develop a practical method for finding such an ,rn. we note from The-orem 6.6 and its corollary that .Lr() y G W' : so (Ax, Ax() tj) in t) for all ./ C F'. Thus, by Lemma 1. we have that (X,A*(AXQ //): 0 for all X G F': that is. A*(Axo y) (). So we need only find a solution .;(l to A*Ax A'y. If, in addition, we assume that rank(.l) - //. then by Lemma 2 we have XQ = (A*A)' A'y. We summarize this discussion in the following theorem. Theorem 6.12. Let A G Mm x(F) and y G F'. Then there exist ./ G F' such that (A*A)x0 = A*I/ and Ax0 - y < A.r - y for all x F' furthermore, if rank(.4) = n. then ,r0 = (A'A)~] A'y. Sec. 6.3 The Adjoint of a Linear Operator 363 To return to our experimenter, let us suppose that the data collected are (1,2), (2, 3), (3, 5), and (1.7). Then / I 1 / 2 A = 2 I 3 1 V4 '7 . r . i 1 2 3 I 1 1 1 1 and y 0 ] 2 I 3 I V4 ' / 3 7 / 30 10 10 ! Thus Therefore 20 V10 30 1 10 2 3 4 d I ' ' 20 V 10 30/ VI /A 3 5 It follows that the line // = 1.7/ is the least squares line. The error computed directly as Axtl y 2 0.3. Suppose that the experimenter chose the times tt (1 < / < m) i piav be satisfy X > = o. Then the two columns of .4 would be orthogonal, so A*A would be a diagonal matrix (see Exercise 19). In this case1, the computations are greatly simplified. In practice, the in x 2 matrix A in our least squares application has rank equal to two. and hence A*A is invertible by the corollary to Lemma 2. For. otherwise, the first column of A is a multiple of the second column, which consists only of ones. But this would occur only if the experimenter collects all the data at exactly one time. Finally, the method above may also be applied if, for some /.:, the ex-perimenter wants to fit a polynomial of degree at most /. to the data. For instance, if a polynomial y el2 + dt + e of degree at most 2 is desired, the appropriate model is ,: - d .'/ -fvA w and A = 364 Chap. 6 Inner Product Spaces Minimal Solutions to Systems of Linear Equations Even when a system of linear equations Ax = b is consistent, there may be no unique solution. In such cases, it may be desirable to find a solution of minimal norm. A solution s to Ax b is called a minimal solution if ||.sj| < ||u|| for all other solutions u. The next theorem assures that every consistent system of linear equations has a unique minimal solution and provides a method for computing it. Theorem 6.13. Let A C Mm x , , (F) and b G F'. Suppose that Ax = b is consistent. Then the following statements are true. (a) There exists exactly one minimal solution s of Ax = b, and s G R(L i-). (b) The vector s is the only solution to Ax = b that lies in R(L,-); that is, if u satisfies (AA*)u - b. then s = A*u. Proof, (a) For simplicity of notation, we let W = R(L4.) and W = N(L.4). Let x be any solution to Ax = b. By Theorem 6.6 (p. 350). ./ = s + y for some s G W and y G W . But W- = W by Exercise 12. and therefore h = Ax = As + Ay = As. So s is a solution to Ax = b that lies in W. To prove (a), we need only show that s is the unique minimal solution. Let u be any solution to Ax b. By Theorem 3.9 (p. 172). we have that v = s + u. where u C W . Since s G W. which equals W by Exercise 12, we have HI2 + > .slh' by Exercise 10 of Section 6.1. Thus s is a minimal solution. We can also see from the preceding calculation that if ||v|| = ||.sj|, then u 0; hence v = s. Therefore s is the unique minimal solution to Ax b. proving (a). (b) Assume that v is also a. solution to Ax = b that lies in W. Then v - , s e w n w ' - w n w - - j ( i | : so V = s. Finally, suppose that (AA*)u = b, and let v = A*u. Then v G W and Av = b. Therefore s v = A*u by the discussion above;. | Example 3 Consider the system x + 2y + z = A x - y + 2z = -U x + 5-1/ = 19. Let Sec. 63 The Adjoint of a Linear Operator 365 To find the minimal solution to this system, we must first find some solution a to AA*x = 6. Now SO we consider the system 6.7- + y+llz = A x + (u/ - Az - -11 llx-4y 4-262= 19, for which one solution is (Any solution will suffice.) Hence s = A*u = is the minima] solution to the given system. EXERCISES 1. Label the following statements as true or false. Assume that the under-lying inner product spaces are finite-dimensional. (a) Every linear operator has an adjoint. (b) Every linear operator on V has the form x > (x.y) for some y G V. (c) For every linear operator T on V and every ordered basis 0 for V, we have [T*]0 = {[T]PY-(d) The adjoint of a linear operator is unique. (e) For any linear operators T and U and scalars a and /;. (aT -f-fcU)* =aV I-6U*. (f) For any n x n matrix .1. we have (L.i)* = L,~. (g) For any linear operator T. we have (T*)* T. 2. Lor each of the following inner product spaces V (over F) and linear transformations g: V /'. find a vector .// such that g(x) = (x.y) for all .r G V. 366 Chap. 6 Inner Product Spaces (a) V = R;i: g(r;.], a2, a-A) Sec. 6.3 The Adjoint of a Linear Operator 367 13. Let T be a linear operator on a finite-dimensional inner product space V. Prove the following results. (a) N(T*T) = N(T). Deduce that rank(T*T) = rank(T). (b) rank(T) = rank(T*). Deduce from (a) that rank(TT*) = rank(T). (c) For any n x n matrix A. rank(A*A) = rank(AA*) = rank(^4). 14. Let V be an inner product space, and let, y. z G V'. Define T: V * V by T(x) (x.y)z for all x G V. First prove that T is linear. Then show that T* exists, and find an explicit expression for it. The following definition is used in Exercises 15-17 and is an extension of the definition of the adjoint of a linear operator. Definition. Let T: V W be a linear transformation, where V and W are finite-dimensional inner product spaces with inner products (, ) j and (, ')2 , respectively. A function T*: W V is called an adjoint of T if (T(x). y)2 = (x, T*(y))j for all x 6 V and y G W. 15. Let T: V + W be a linear transformation, where V and W are finite-dimensional inner product spaces with inner products {, ), and {, ),,, respectively. Prove the following results. (a) There is a unique adjoint T* of T, and T* is linear. (b) If 3 and 7 are orthonormal bases for V and W. respectively, then [T1? = ([T]}Y-(c) rank(T*) = rank(T). (d) (T*(aO,y)j = (x, T(y))2 for all x G W and y G V. (e) For all x G V. T*T(>) = 0 if and only if T(x) = 0. 16. State and prove a result that extends the first four parts of Theorem 6.11 using the preceding definition. 17. Let T: V * W be a linear transformation, where V and W are finite-dimensional inner product spaces. Prove that (R(T*)) ' = N(T), using the preceding definition. 18.T Let A be an n x n matrix. Prove that det(.4*) = det(A). 19. Suppose that A is an rnxn matrix in which no two columns are identical. Prove that A* A is a diagonal matrix if and only if every pair of columns of A is orthogonal. 20. For each of the sets of data that follows, use the least squares approx-imation to find the best fits with both (i) a linear function and (ii) a quadratic function. Compute the error E in both cases. (a) {(-3,9), ( -2 ,6) , (0,2), (1,1)} 368 Chap. 6 Inner Product Spaces (b) {(1,2), (3,4), (5,7), (7,9), (9,12)} (c) {(-2,4), ( -1 ,3) , (0,1), (1,-1) , (2 , -3)} 21. In physics. Hooka's law states that (within certain limits) there is a linear relationship between the length x of a spring and the force y applied to (or exerted by) the spring. That is, y = ex + d, where c is called the spring constant . Use the following data to estimate the spring constant (the length is given in inches and the force is given in pounds). Length X 3.5 4.0 4.5 5.0 Force y 1.0 2.2 2.8 4.3 22. Find the minimal solution to each of the following systems of linear (xiuations. (aj x + 2y - z = 12 x+y-z=0 (c) 2x -y + z = 3 x-y+z=2 x + 2y - z = 1 (b) 2x + 3y + 2 = 2 Ax + ly - z = 4 (d) x + y + z w, 1 2x - y w = 1 23. Consider the problem of finding the least squares line y = ct + d corre-sponding to the rn observations (t, ij), (f2, y2),..., (tm,ym). (a) Show that the equation (A*A)xo = A*y of Theorem 6.12 takes the form of the normal equations: ' ? < ' ti)d = Y^tiVi and ^ L c + rnd-- ^2'yi-,i=1 i-1 These equations may also be obtained from the error E by setting the partial derivatives of E with respect to both c and d. equal to zero. Sec. 6.4 Normal and Self-Adjoint Operators 369 (b) Use the second normal equation of (a) to show that the least squares line must pass through the center of mass, (i,y), where - 1 V^ tnd y = m i=l 24. Let V and {ei, e 2 , . . . } be defined as in Exercise 23 of Section 6.2. Define T: V -* V by oo T(a)(k) = y a(i) for every positive integer A:. i=k Notice that the infinite series in the definition of T converges because a(i) ^ 0 for only finitely many i. (a) Prove that T is a linear operator on V. (b) Prove that for any positive integer n, T(en) = Yl7=i ei-(c) Prove that T has no adjoint. Hint: By way of contradiction, suppose that T* exists. Prove that for any positive integer n, T*(en)(k) 7^ 0 for infinitely many k. 6.4 NORMAL AND SELF-ADJOINT OPERATORS We have seen the importance of diagonalizable operators in Chapter 5. For these operators, it is necessary and sufficient for the vector space V to possess a basis of eigenvectors. As V is an inner product space in this chapter, it is reasonable to seek conditions that guarantee that V has an orthonormal basis of eigenvectors. A very important result that helps achieve our goal is Schur's theorem (Theorem 6.14). The formulation that follows is in terms of linear operators. The next section contains the more familiar matrix form. We begin with a lemma. Lemma. Let T be a linear operator on a finite-dimensional inner prodtict space V. If T has an eigenvector, then so does T*. Proof Suppose that e is an eigenvector of T with corresponding eigenvalue A. Then for any x G V, 0 = (0.x) = ((T - W)(v),x) = (v. (T - Al)*(ar)> = (v, (T* - Al)(ar)> , and hence v is orthogonal to the range of T* Al. So T* AI is not onto and hence is not one-to-one. Thus T* Al has a nonzero null space, and any nonzero vector in this null space is an eigenvector of T* with corresponding eigenvalue A. 370 Chap. 6 Inner Product Spaces Recall (see the exercises of Section 2.1 and see Section 5.4) that a subspace W of V is said to be T-invariant if T(W) is contained in W. If W is T~ invariant, we may define the restriction Tw: W W by Tw(x) T(x) for all x G W. It is clear that Tw is a linear operator on W. Recall from Section 5.2 that a polynomial is said to split if it factors into linear polynomials. Theorem 6.14 (Schur). Let T be a linear operator on a finite-dimensional inner product space V. .Suppose that the characteristic poly-nomial of T splits. Then there exists an orthonormal basis 0 forV such that the matrix [T].-< is upper triangular. Proof. The proof is by mathematical induction on the dimension // of V. The result is immediate if n I. So suppose that the result is true for linear operators on (it . l)-dimensional inner product spaces whose characteristic polynomials split. By the lemma, we can assume that T* has a unit eigen-vector 2. Suppose that T*(z) = Xz and that W - span({~}). We show that W ' is T-invariant. If y W ' and x ~ cz G W, then M (T(y),x> = =_Sec. 6.4 Normal and Self-Adjoint Operators 371 Example 1 Let T: R- * R2 be rotation by 0. where ()372 Chap. 6 Inner Product Spaces = (xi, X2x2) = X2 (xi,x2) Since Ai ^ A2, we conclude that (x,x2) = 0 . 1 Theorem 6.16. Let T be a linear operator on a hnitc-dimensional com-plex inner product space V. Then T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T. Proof. Suppose that T is normal. By the fundamental theorem of algebra (Theorem D.4), the characteristic polynomial of T splits. So we may apply Schur's theorem to obtain an orthonormal basis f3 = {v,v2,... ,vn} for V such that [T]tf = A is upper triangular. We know that V] is an eigenvector of T because A is upper triangular. Assume that V, v2,... ,Vk-i are eigen-vectors of T. We claim that vk is also an eigenvector of T. It then follows by mathematical induction on k that all of the Uj's are eigenvectors of T. Consider any j < k, and let A; denote the eigenvalue of T corresponding to Jo-By Theorem 6.15, T*(VJ) = XjVj. Since A is upper triangular. T(vk) = Akvi + A2kv2 + + AjkVj + + Akkvk. Furthermore, by the corollary to Theorem 6.5 (p. 347), Ajk = (T(vk),Vj) = (vk,T*(Vj)) = (vk,XjVj) = A, (vkfVj) = 0. It follows that T(vk) = Akkvk, and hence vk is an eigenvector of T. So by induction, all the vectors in p* are eigenvectors of T. The converse was already proved on page 370. I Interestingly, as the next example shows, Theorem 6.16 does not extend to infinite-dimensional complex inner product spaces. Example 3 Consider the inner product space H with the orthonormal set S from Exam-ple 9 in Section 6.1. Let V = span(S'), and let T and U be the linear operators on V defined by T( / ) - / , / and U(/) = f_uf. Then T( / n ) = /.+ ! and U(/n) = / n _ ] for all integers n. Thus (T ( / m ) , / n ) = (/m+li/n) = Sec. 6.4 Normal and Self-Adjoint Operators 373 Hence J^aifi+1=T(f) = Xf = TXaifi Since am ^ 0, we can write / m +i as a linear combination of f, / n +i , ,fm-But this is a contradiction because S is linearly independent. Example 1 illustrates that normality is not sufficient to guarantee the existence of an orthonormal basis of eigenvectors for real inner product spaces. For real inner product spaces, we must replace normality by the stronger condition that T = T* in order to guarantee such a basis. Definitions. Let T be a linear operator on an inner product space V. We say that. T is self-adjoint (Hermitian) if T = T*. An n x n real or complex matrix A is self-adjoint (Hermitian) if A = A*. It follows immediately that if 0 is an orthonormal basis, then T is self-adjoint if and only if [T]^ is self-adjoint, for real matrices, this condition reduces to the requirement that A be symmetric. Before we state our main result for self-adjoint operators, we need some preliminary work. By definition, a linear operator on a real inner product space has only real eigenvalues. The lemma that follows shows that the same can *be said for self-adjoint operators on a complex inner product space. Similarly, the characteristic polynomial of every linear operator on a complex inner product space splits, and the same is true for self-adjoint operators on a real inner product space. Lemma. Let T be a self-adjoint operator on a finite-dimensional inner prochict space V. Then (a) Every eigenvalue o/'T is real. (b) Suppose that V is a real inner product space. Then the characteristic polynomial of T splits. Proof, (a) Suppose that T(x) = Xx for x -/= 0. Because a self-adjoint operator is also normal, we can apply Theorem 6.f 5(c) to obtain XX = T(X) = T*(X) = X. So A = A; that is. A is real. (b) Let n. dim(V), 0 be an orthonormal basis for V, and A = [T]p. Then A is self-adjoint. Let T A be the linear operator on C' defined by TA(X) = Ax for all x G C'. Note that TA is self-adjoint because [T.4]7 = A, where 7 is the standard ordered (orthonormal) basis for C'. So, by (a), the eigenvalues of ~T A are real. By the fundamental theorem of algebra, the 374 Chap. 6 Inner Product Spaces characteristic polynomial of T.4 splits into factors of the form t A. Since each A is real, the characteristic polynomial splits over R. But T.4 has the same characteristic polynomial as A, which has the same characteristic polynomial as T. Therefore the characteristic polynomial of T splits. I We are now able to establish one of the major results of this chapter. Theorem 6.17. Let T be a linear operator on a finite-dimensional real inner product space V. Then T is self-adjoint if and only if there exists an orthonormal basis 0 for V consisting of eigenvectors of T. Proof. Suppose that T is self-adjoint. By the lemma, we may apply Schur's theorem to obtain an orthonormal basis 0 for V such that the matrix A = T]p is upper triangular. But A* = [T]J = [Tfc = [T)p = A. So A and A* are both upper triangular, and therefore A is a diagonal matrix. Thus 3 must consist of eigenvectors of T. The converse is left as an exercise. I Theorem 6.17 is used extensively in many areas of mathematics and statis-tics. We restate this theorem in matrix form in the next section. Example 4 As we noted earlier, real symmetric matrices are self-adjoint, and self-adjoint matrices are normal. The following matrix A is complex and symmetric: * - ( ! l ) -* A*={-i 1 But A is not normal, because (AA*)2 = 1 + i and (A*A)2 = 1 i. Therefore complex symmetric matrices need not. be normal. EXERCISES 1. Label the following statements as true or false. Assume that the under-lying inner product spaces are finite-dimensional. (a) Every self-adjoint operator is normal. (b) Operators and their adjoints have the same eigenvectors. (c) If T is an operator on an inner product space V. then T is normal if and only if T]p is normal, where 0 is any ordered basis for V. (d) A real or complex matrix A is normal if and only if L.4 is normal. (e) The eigenvalues of a self-adjoint operator must all be real. Sec. 6.4 Normal and Self-Adjoint Operators 375 (f) The identity and zero operators are self-adjoint. (g) Every normal operator is diagonalizable. (h) Every self-adjoint operator is diagonalizable. 2. For each linear operator T on an inner product space V, determine whether T is normal, self-adjoint, or neither. If possible, produce an orthonormal basis of eigenvectors of T for V and list the corresponding eigenvalues. (a) V - R2 and T is defined by T(a.b) = (2a - 26, -2a 56). (b) V - R:i and T is defined by T(a, 6, c) = ( - a + b, 56, Aa - 26 + 5c). (c) V - C2 and T is defined by T(o, 6) = (2a + ib. a + 26). (d) V - P2(/r) and T is defined by T( / ) - / ' . where (/,= / f(t)g(t)dt. (e) V = M2x2(/?) and T is defined by T(A) =. A< (f) V = M2x2(i?) and T is defined by T d c d a 6 3. Give an example of a linear operator T on R2 and an ordered basis for R that provides a counterexample to the statement in Exercise 1(c). > 4. Lei T and U be self-adjoint operators on an inner product space V. Prove that TU is self-adjoint if and only if TU = UT. 5. Prove (b) of Theorem 6.15. 6. Let V be a complex inner product space, and let T be a linear operator on V. Define Ti = i ( T + r ) and T2 = i ( T - T*). 2 2/ (a) Prove that T| and T2 are self-adjoint and that T = Tj +iT2. (b) Suppose also that T = U,i +iU2 , where U| and U2 are self-adjoint. Prove that Ui = T] and U2 = T2. (c) Prove that T is normal if and only if T |T 2 = T 2 T| . 7. Let T be a linear operator on an inner product space V, and let W be a T-invariant subspace of V. Prove the following results. (a) If T is self-adjoint, then Tw is self-adjoint. (b) W is T'-invariant. (c) If W is both T- and T*-invariant, then (Tw)* = 0~*)w (d) If W is both T- and T*-invariant and T is normal, then Tw is normal. r 376 Chap. 6 Inner Product Spaces 8. Let T be a normal operator on a finite-dimensional complex inner product, space V. and let W be a. subspace of V. Prove that if W is T-invariant, then W is also T*-invariant. Hint: Use Exercise 24 of Sec-tion 5.4. 9. Let T be a normal operator on a finite-dimensional inner product space V. Prove that N(T) - N(T*) and R(T) - R(T*). Hint: Use Theo-rem 6.15 and Exercise 12 of Section 6.3. 10. Let. T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x C V ! |T(;r); ;r | |2-! |T0r) | |2 + ||^||2. Deduce that T - , is invertible and that [(T - /I)'1]* = (T + i)~l. 11. Assume that T is a. linear operator on a complex (not necessarily finite-dimensional) inner product space V with an adjoint T*. Prove the following results. (a) If T is self-adjoint, then (T(x),x) is real for all ./ G V. (b) If T satisfies (T(x),x) = 0 for all x G V. then T = T. Hint: Replace x by x I y and then by x f iy, and expand the resulting inner products. (c) If (T(x).x) is real for all x G V. then T = T*. 12. Let T be a normal operator on a finite-dimensional real inner product space V whose characteristic polynomial splits. Prove that V has an orthonormal basis of eigenvectors of T. Hence prove that T is self-adjoint. 13. An n. x n. n>al matrix A is said to be a Gramian matrix if there exists a real (square) matrix 13 such that A P' B. Prove that A is a Gramian matrix if and only if A is symmetric and all of its eigenvalues are non-negative. Hint: Apply Theorem 6.17 to T = L4 to obtain an orthonor-mal basis {('], y 2 , . . . . v} of eigenvectors with the associated eigenvalues Ai, A2,. . ., A. Define the linear operator U by U (*';) = /X~iV;. 14. Simultaneous Diagonalization. Let V be a finite-dimensional real inner product space, and let U and T be self-adjoint linear operators on V such that UT = TU. Prove that there exists an orthonormal basis for V consisting of vectors that are eigenvectors of both U and T. (The complex version of this result appears as Exercise 10 of Section 6.6.) Hint: For any eigenspace W E^ of T. we have that W is both T- and U-invariant. By Exercise 7. we have that W L is both T- and U-invariant. Apply Theorem 6.17 and Theorem 6.6 (p. 350). Sec. 6.4 [Jormal and Self-Adjoint Operators 377 15. Let .4 and B be symmetric n x n matrices such that AB = BA. Use Exercise 14 to prove that there exists an orthogonal matrix P such that P'AP and PtBP an- both diagonal matrices. 16. Prove the Cayley Hamilton theorem for a complex n.xn matrix A. That is. if /'(/) is the characteristic polynomial of A, prove that f{A) (). Hint: Use Schur's theorem to show that A may be assumed to be upper triangular, in which case /(*) = U(Au -t) i i Now if T = l_4. we have (Ajj - T)(e ;) G span({< h e 2 e, i}) for j > 2, where {e.,e2,..., en} is the standard ordered basis for C'. (The general case is proved in Section 5.4.) The following definitions are used in Exercises 17 through 23. Definitions. A linear operator T on a. linite-dimeusional inner product space is called positive definite [positive scmidefinitc] if T is self-adjoint and (T(x),x) > 0 [(T(x)..x) > 0] for all x f 0. An ii x n matrix A with entries from R or (' is called positive definite positive semidefinite} HL, is positive definite [positive semideBmte]. 17. Let T and U be a self-adjoint linear operators on an //-dimensional inner product space V. and let A = [T]^, where 0 is an orthonormal basis for V. Prove the following results. (a) T is positive definite [semidefinite] if and only if all of its eigenval-ues are positive [nonnegative]. (b) T is positive definite if and only if y AjjO.jU.j > 0 for all nonzero //.-tuples (a i ,o 2 , . . . ,an) . (c) T is positive semidefinite if and only if A B* 11 for some square matrix II. (d) If T and U are positive semidefinite operators such that T2 U2. then T = U. (e) If T and U are positive definite operators such that TU = UT, then TU is positive definite. (f) T is positive definite [semidefinite] if and only if .4 is positive def-inite [semidefinite]. Because of (f). results analogous to items (a) through (d) hold for ma-trices as well as operators. 378 Chap. 6 Inner Product Spaces 18. Let T: V > W be a linear transformation, where V and W are finite-dimensional inner product spaces. Prove the following results. (a) T*T and TT* are positive semidefinite. (See Exercise 15 of Sec-tion 6.3.) (b) rank(T*T) = rank(TT*) = rank(T). 19. Let T and U be positive definite operators on an inner product space V. Prove the following results. (a) T + U is positive definite. (b) If c > 0, then cT is positive definite. (c) T is positive definite. 20. Let V be an inner product space with inner product (, ) anSec. 6.5 Unitary and Orthogonal Operators and Their Matrices 379 24. This argument gives another proof of Schur's theorem. Let T be a linear opera!or on a finite dimensional inner product space V. (a) Suppose that 0 is an ordered basis for V such that [T],^ is an upper triangular matrix. Let 7 be the orthonormal basis for V obtained by applying the Gram Schmidt orthogonalization process to 0 and then normalizing the resulting vectors. Prove that [T]7 is an upper triangular matrix. (b) Use Exercise 32 of Section 5.4 and (a) to obtain an alternate proof of Schur's theorem. 6.5 UNITARY AND ORTHOGONAL OPERATORS AND THEIR MATRICES In this section, we continue our analogy between complex numbers and linear operators. Recall that the adjoint of a. linear operator acts similarly to the conjugate of a complex number (see, for example, Theorem 6.11 p. 359). A complex number 2 has length 1 if zz = 1. In this section, we study those lineai- operators T on an inner product space V such that TT* = T*T I. We will see that these are precisely the linear operators that 'preserve length' in the sense that ||T(x)|| = ||x|| for all x G V. As another characterization, we prove that, on a finite-dimensional complex inner product space, these are the normal operators whose eigenvalues all have absolute value I. In past chapters, we were interested in studying those functions that pre-serve the structure of the underlying space. In particular, linear operators preserve the operations of vector addition and scalar multiplication, and iso-morphisms preserve all the vector space structure. It is now natural to con-sider those linear operators T on an inner product space that preserve length. We will see that this condition guarantees, in fact, that T preserves the inner product. Definitions. Let T be a linear operator on a Unite-dimensional inner product space V (over F). If ||T(.z-)j| = ||./;|| for all x G V, we cail T a unitary operator if F (' and an orthogonal operator if F R. It should be noted that, in the infinite-dimensional case, an operator sat-isfying the preceding norm requirement is generally called an isornetry. If, in addition, the operator is onto (the condition guarantees one-to-one), then the operator is called a unitary or orthogonal operator. Clearly, any rotation or reflection in R2 preserves length and hence is an orthogonal operator. We study these operators in much more detail in Section 6.1 I. 380 Chap. 6 Inner Product Spaces Example 1 Let h G H satisfy h(x) = 1 for all x. Define the linear operator T on H by T( / ) = hf. Then l|T(/)||2 = hff = -J 'h( t ) f ( t )Mjm)dt = Wff since |/?(/)|2 = 1 for all t. So T is a unitary operator. Theorem 6.18. Let T be a linear operator on a finite-dimensional inner product space V. Then the following statements are equivalent. (a) TT* = T*T = I. (b) (T(x),T(y)) = (x.y) for all x,y G V. (c) If ft is an orthonormal basis for V, then T(0) is an orthonormal basis for V. (d) There exists an orthonormal basis 0 for V such that. T(/3) is an orthonor-mal basis for V. (e) ||T(:/;)|| = ||z|| for all x V. Thus all the conditions above are equivalent to the definition of a uni-tary or orthogonal operator. From (a), it follows that unitary or orthogonal operators are normal. Before proving the theorem, we first prove a lemma. Compare this lemma to Exercise 11(b) of Section 6.4. Lemma. Let U be a self-adjoint operator on a finite-dimensional inner product space V. If (x, U(ar)) = 0 for all x G V, then U = T0 . Proof By either Theorem 6.16 (p. 372) or 6.17 (p. 374). we may choose an orthonormal basis 0 for V consisting of eigenvectors of U. If x G 0, then U(x) = Xx for some A. Thus 0 = (x. U(.r)> = Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices for some scalars a,;, and so 381 Ixll2 = ^2aiVi,J2aJvJ 3=1 a ii i=i j=i i=i j= E i . i2 i= since 0 is orthonormal. Applying the same manipulations to n T(x) = ]T>T( - , 0 r 1 and using the fact, that T(0) is also orthonormal, we obtain I I T M I P = D , I ' - ' . i - i Hence ||T(.r)|| = ||.x||. Finally, we prove? that (e) implies (a). For any x G V, we have [X.X) x = T(x)2 = (T(x),T(x)) = (x,T*T(x) So (re, (I -T*T)0r)) - 0 for all .; G V. Let U = I - T*T; then U is self-adjoint, and (re, U(x)) = 0 for all x G V. Hence, by the lemma, we have To = U = I T*T. and therefore T*T I. Since V is finite-dimensional, we may use Exercise 10 of Section 2.4 to conclude that TT* I. It follows immediately from the definition that every eigenvalue of a uni-tary or orthogonal operator has absolute value 1. In fact, even more is true. Corollary 1. Let T be a linear operator on a finite-dimensional real inner product space V. Then V has an orthonormal basis of eigenvectors of T with corresponding eigenvalues of absolute value 1 if and only ifT is both sell-adjoint and orthogonal. Proof. Suppose that V has an orthonormal basis {?,'|, v2, , vn} such that T(vi) Xji'i and |A,| = 1 for all i. By Theorem 6.17 (p. 374), T is self-adjoint. Thus (TT*)(vi) = T(Xii'i) = A;A;e, = X'jvj = Vi for each /'. So TT* = I, and again by Exercise 10 of Section 2.4. T is orthogonal by Theorem 6.18(a). If T is self-adjoint, then, by Theorem 6.17, we have that V possesses an orthonormal basis {v, t ' 2 , . . . , vn} such that T(v.i) = A,.v.j for all ?'. If T is also orthogonal, we have I'M 'INI = IIAjUiH = ||T(UJ)|| = vi; so I Aj I = 1 for every i. 382 Chap. 6 Inner Product Spaces Corollary 2. Let T be a linear operator on a finite-dimensional complex inner product space V. Then V has an orthonormal basis of eigenvectors ofT with corresponding eigenvalues of absolute value 1 if and only ifT is unitary: Proof. The proof is similar to the proof of Corollary 1. j| Example 2 Let T: R2 > R2 be a rotation by 6>, where 0 < 0 < n. It is clear geometrically that T 'preserves length,' that is, that ||T(:r)|| = x for all x G R2. The fact that rotations by a fixed angle preserve perpendicularity not only can be seen geometrically but now follows from (b) of Theorem 6.18. Perhaps the fact that such a transformation preserves the inner product is not so obvious: however, we obtain this fact from (b) also. Finally, an inspection of the matrix representation of T with respect to the standard ordered basis, wThich is cos 0 sin i sin 6 cos i reveals that T is not self-adjoint for the given restriction on 9. As we men-tioned earlier, this fact also follows from the geometric observation that T has no eigenvectors and from Theorem 6.15 (p. 371). It is seen easily from the preceding matrix that Tx is the rotation by $. Definition. Let L be a one-dimensional subspace of R2. We may view L as a line in the plane through the origin. A linear operator T on R2 is called a reflection of R2 about L if T(x) = x for all x G L and T(x) x for all As an example of a reflection, consider the operator defined in Example 3 of Section 2.5. Example 3 Let T be a reflection of R2 about a line L through the origin. We show that T is an orthogonal operator. Select vectors v G L and v2 G L1- such that vi || v2 = 1- Then T(?;i) = v and T(v2) v2. Thus v and v2 are eigenvectors of T with corresponding eigenvalues 1 and 1. respectively. Furthermore, {vi,v2} is an orthonormal basis for R2. It follows that T is an orthogonal operator by Corollary 1 to Theorem 6.18. We now examine the matrices that represent unitary and orthogonal trans-formations. Definitions. A square matrix A is called an orthogonal matrix if A* A = AAl = I and unitary if A* A = A A* = I. Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 383 Since for a real matrix A we have A* = A1, a real unitary matrix is also orthogonal. In this case, we call A orthogonal rather than unitary. Note that the condition AA* = / is equivalent to the statement that the rows of A form an orthonormal basis for Fn because Si, = hj = {AA*)ij = J2Aik(A*)kj = Y^AikAjk, k- k=i and the last term represents the inner product of the ith and jfth rows of A. A similar remark can be made about the columns of A and the condition A* A = I. It also follows from the definition above and from Theorem 6.10 (p. 359) that a linear operator T on an inner product space V is unitary [orthogonal] if and only if [T]p is unitary [orthogonal] for some orthonormal basis 0 for V. Example 4 From Example 2, the matrix cos 9 sin 9 sin 9 cos 9 is clearly orthogonal. One can easily see that the rows of the matrix form an orthonormal basis for R . Similarly, the columns of the matrix form an orthonormal basis for R2. v ., Example 5 Let T be a reflection of R2 about a line L through the origin, let 0 be the standard ordered basis for R2, and let A = [T]g. Then T = L^. Since T is an orthogonal operator and 0 is an orthonormal basis, A is an orthogonal matrix. We describe A. Suppose that a is the angle from the positive x-sxis to L. Let v (cos a, sin a) and t'2 = (sin a, cos a). Then ||i;i|| = ||v2|| = 1, v G L, and v2 G L1 . Hence 7 = {v,v2} is an orthonormal basis for R2. Because T(fi) = v and T(v2) = v2, we have [T]7 = [lA]y = Let Q = cos a sin a sin a cos a ly the corollary to Theorem 2.23 (p. 115), A = Q[lA1Q-1 384 Chap. 6 Inner Product Spaces cos a sin o shm f 0 cos a J l o - I COSQ sine* sin a COSQ cos- a sin a j sin a coso 2sinacosa (cos o. sin of) I sin 2o We know that, for a complex normal real symmetric matrix .4. there exists an orthonormal basis Q for F' consisting of eigenvectors of A. Hence .4 is similar to a diagonal matrix D. By the corollary to Theorem 2.23 (p. 115), the matrix Q whose columns are the vectors in 3 is such that D Q AC). But since t he columns of Q are an orl lionoiinal basis for F'. it follows that Q is unitary [orthogonal . In this case, we say that A is unitarily equivalent orthogonally equivalent, to D. It is easily seen (see Exercise 18) that this relation is an equivalence relation on M.(C) [M, (li). More generally. .4 and II are unitarily equivalent [orthogonally equivalent] if and only if there exists a unitary [orthogonal] matrix P such thai A P' IIP. The preceding paragraph has proved half of each of the next two theo-rems. Theorem 6.19. Let A be a complex n x n matrix. Then A is normal if and only if A is unitarily equivalent to a diagonal matrix. Proof. By the preceding remarks, we need only prove that if A is unitarily equivalent to a diagonal matrix, then A is normal. Suppose that A P*DP. where P is a unitary matrix and I) is a diagonal matrix. Then AA* = (P*DP)(P*DP)* - (P*DP){P*D*P) P*DID*P P*DD*P. Similarly. A*A = P*D*DP. Since D is a diagonal matrix, however, we have DD* =D*D. Thus AA* = A*A. I Theorem 6.20. Let A be a real n x /; matrix. Then A is symmetric if and only if A is orthogonally equivalent to a real diagonal matrix. Proof. The proof is similar to the proof of Theorem 6.19 and is left as an exercise. 1 Example 6 Let Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 385 Since A is symmetric, Theorem 6.20 tells us that A is orthogonally equivalent to a diagonal matrix. We find an orthogonal matrix P and a diagonal matrix D such that PtAP= I). To find P. we obtain an orthonormal basis of eigenvectors. It is easy to show that the eigenvalues of .4 are 2 and 8. The set {(-1,1,0), ( -1,0, 1)} is a basis for the eigenspace corresponding to 2. Because this set is not orthogonal, we apply the Gram Schmidt process to obtain the orthogonal set {(-1. 1.0). -1(1,1 . -2)}. The set {(1.1.1)} is a basis for the eigenspace corresponding to 8. Notice that (1,1,1) is orthogonal to the preceding two vectors, as predicted by Theorem 6.15(d) (p. 371). Taking the union of these two bases and normalizing the vectors, we obtain the following orthonormal basis for R'5 consisting of eigenvectors of A: 4 ( 1 ,1 ,0 ) , - ^ (1 ,1 , -2 ) , - U l , l , l ) ) . V2 v0 v3 J Thus one possible choice for P is I 1 p = 0 i 2 /6 I v ':; l v/5/ '2 0 if I 0 2 0 .0 0 8 Because of Schur's theorem (Theorem 6.14 p. 370). the next 'result is immediate. As it is the matrix form of Schur's theorem, we also refer to it as Schur's theorem. Theorem 6.21 (Schur). Let A C MX(F) be a matrix whose charac-teristic polynomial splits over F. (a) If F = ('. then A is unitarily equivalent to a complex upper triangular matrix. (b) If F R. then A is orthogonally equivalent to a real upper triangular matrix. Rigid Motions* The purpose of this application is to characterize the so-called rigid mo-tions of a finite-dimensional real inner product space. One may think intu-itively of such a motion as a transformation that does not affect the shape of a figure under ifs action, hence the term rigid. The key requirement for such a transformation is that it preserves distances. Definition. Let V be a real inner product space. A function f: V * V is called a rigid motion if !./(') -f(y) = x-y 386 Chap. 6 Inner Product Spaces for all x,y G V. For example, any orthogonal operator on a finite-dimensional real inner product space is a rigid motion. Another class of rigid motions are the translations. A function g: V V, where V is a real inner product, space, is called a t ranslat ion if there exists a vector e G V such that g(x) = X + VQ for all x G V. We say that g is the translation by VQ. It is a simple exercise to show that translations, as well as composites of rigid motions on a real inner product space, are also rigid motions. (See Exercise 22.) Thus an orthogonal operator on a finite-dimensional real inner product space V followed by a translation on V is a rigid motion on V. Remarkably, every rigid motion on V may be characterized in this way. Theorem 6.22. Let f: V V be a rigid motion on a Hnite-dimcnsional real inner product space V. Then there exists a unique orthogonal operator T on V and a unique translation g on V such that f = g o T. Any orthogonal operator is a special case of this composite, in which the translation is by 0. Any translation is also a special case, in which the orthogonal operator is the identity operator. Proof. Let T: V V be defined by T{x)=f(x)-f{0) for all x G V. We show that T is an orthogonal operator, from which it follows that / = g o T, where g is the translation by f(0). Observe that T is the composite of / and the translation by f(0); hence T is a rigid motion. Furthermore, for any x G V ||T(x)||2 = !|/(.r) -f(0)2 = x-0[2 = x2. and consequently ||T(x)|| = ||x|| for any x G V. Thus for any x. y G V. ||T(.r) - T(y)2 = ||T(z)||2 - 2 (T(x).T(y)) + ||T(y)||2 2Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 387 = ||T(ar + ay) - T(x)2 + a2T(y)2 - 2a (T(x + ay) - T(x), T(y)> = (x + ay) - x2 + a2yf - 2a[{T(x + oy),T(y)> - (T(z),T(y))] = a2y2 + a2y[2 - 2a[(x + ay,y) - (x,y)] = 2a2[y2-2a[(x,y)+ay2-(x,y)] = 0. Thus T(x -f- ay) = T(.r) + oT(y), and hence T is linear. Since T also preserves inner products. T is an orthogonal operator. To prove uniqueness, suppose that un and VQ are in V and T and U are orthogonal operators on V such that f(x)=T(x)+u0 = U(x)+v0 for all x G V. Substituting x 0 in the preceding equation yields WQ = Vn, and hence the translation is unique. This equation, therefore, reduces to T(x) = U(.r) for all x G V. and hence T -- U. I Orthogonal Operators on R2 Because of Theorem 6.22, an understanding of rigid motions requires a characterization of orthogonal operators. The next, result characterizes or-thogonal operators on R2. We postpone the case of orthogonal operators on more general spaces to Section 6.1 1. Theorem 6.23. Let. T be an orthogonal operator on R , and let A = [T]^, where 0 is the standard ordered basis for R2. Then exactly one of the following conditions is satisfied: (a) T is a rotation, and det(A) = 1. (b) T is a reflection about a line through the origin, and det(A) = 1. Proof. Because T is an orthogonal operator, T(,tf) = {T(ei),T(e2)} is an orthonormal basis for R2 by Theorem 6.18(c). Since T(ei) is a unit vector, there is a unique angle 9. 0 < 9 < 2TT, such that T(ei) = (cos9,sin9). Since T(e2) is a unit vector and is orthogonal to T(ei), there are only two possible choices for T(c2). Either T(e2) = (-sin0,cos0) or T(e2) = (sin0, -cos6>). r,. . _ , _ , , , . , .. . (e.os9 sin i first, suppose that I (e->) - s i n 9, cos 0). 1 hen ,4 I . ., v ~' K ' s9 cosi It follows from Example 1 of Section 6.4 that T is a rotation by the angle 9. Also del.(.4) = cos2 0 + sin2 9 = 1. 388 Now suppose that T(e2) = (sin0, cos0). Then A = Chap. 6 Inner Product Spaces cos 9 sin 0' sin0 cos0 Comparing this matrix to the matrix A of Example 5, we see that T is the reflection of R2 about a line L, so that a = 0/2 is the angle from the positive x-axis to L. Furthermore, det(A) = - c o s 2 0 - s i n 2 0 = - l . | Combining Theorems 6.22 and 6.23, we obtain the following characteriza-tion of rigid motions on R2. Corollary. Any rigid motion on R2 is either a rotation followed by a trans-lation or a reflection about a line through the origin followed by a translation. Example 7 Let A = 2 /5 - 1 7 5 We showjthat LA is the reflection of R2 about a line L through the origin, and then describe L. Clearly AA* = A* A = I, and therefore A is an orthogonal matrix. Hence LA is an orthogonal operator. Furthermore, 1 4 det(>4 = - - - - = -1 , 5 5 and thus L.4 is a reflection of R2 about a line L through the origin by The-orem 6.23. Since L is the one-dimensional eigenspace corresponding to the eigenvalue I of L4, it suffices to find an eigenvector of L4 corresponding to 1. One such vector is v = (2, /5 1). Thus L is the span of {?;}. Alternatively. L is the line through the origin with slope (/5 l)/2, and hence is the line with the equation ' 5 - 1 y = Conic Sections As an application of Theorem 6.20, we consider the quadratic equation ax2 + 2bxy + ay2 + dx + ey + f = 0. (2) Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 389 For special choices of the coefficients in (2), we obtain the various conic sections. For example, if a = c = 1, b = d = e = 0, and / = 1, we obtain the circle x 4- y2 = 1 with center at the origin. The remaining conic sections, namely, the ellipse, parabola, and hyperbola, are obtained by other choices of the coefficients. If /> = 0, then it is easy to graph the equation by the method of completing the square because the xy-term is absent. For example, the equation x2 + 2x + y2 + Ay + 2 0 may be rewritten as (x + I)2 4- (y + 2)2 = 3, which describes a circle with radius A/3 and center at (1,-2) in the ,r-t/-coordinate system. If we consider the transformation of coordinates (x.y) (x'.y'). where x' = x + 1 and y' = y + 2, then our equation simplifies to ' ,,.l2 IV '2 3. This change of variable allows us to eliminate the x- and y-terms. We now concentrate solely on the elimination of the xy-term. To accom-plish this, we consider the expression ax2 + 2bxy + cy2, (3) which is called the associated quadratic form of (2). Quadratic forms are studied in more generality in Section 6.8. If we let A = a b b e and X then (3) may be written as X'AX = (AX.X). For example, the quadratic form 3;r2 + Axy + 6y may be written as The fact that A is symmetric is crucial in our discussion. For, by Theo-rem 6.20, we may choose an orthogonal matrix P and a diagonal matrix D with real diagonal entries Ai and A2 such that P'AP = D. Now define X' = by X' = P'X or, equivalently, by PX' = PT>fX = X. Then XlAX = (PX')tA(PXl) = X'f(PtAP)X' = X'DX' = Xx{x')2 XiilJ >2 Thus the transformation (x.y) * (x'.y') allows us to eliminate the xy-term in (3), and hence in (2). Furthermore, since P is orthogonal, we have by Theorem 6.23 (with T = Lp) that det(P) = 1. If det(P) = 1, we may interchange the columns 390 Chap. 6 Inner Product Spaces of P to obtain a matrix Q. Because the columns of / ' form an orthonormal basis of eigenvectors of .1. the same is true of the columns of Q. Therefore, Q'AQ- A2 0 0 A, Notice that de(Q) -det(P) = 1. So, if det(P) - 1. we can take Q for our new /'; consequently, we may always choose P so that det(P) = I. By Lemma4 to Theorem 6.22 (with T = L/-)- ' follows that matrix P represents a rotation. In summary, the .///-term in (2) may be eliminated by a rotation of the x-axis and //-axis to new axes x' and //' given by X PX'. where P is an orthogonal matrix and del (P) 1. Furl hermore, the coefficients of (./')-' and (//') are the eigenvalues of A = a b b e This result is a. restatement of a result known as t he principal axis theort m for R2. The arguments above, of course, are easily extended to quadratic ((Illations in n variables. For example, in the case // - .3. by special choices of the coefficients, we obtain the quadratic surfaces the elliptic cone, the ellipsoid! the hyperbolic paraboloid, etc. As an illustration of the preceding transformation, consider the quadratic equation 2x2 xy + ryy2 - 3 6 = 0, for which the associated quadratic form is 2x Axy I 5//2. In the notation we have been using, 4 2 - 2 - 9 n so that the eigenvalues of A are 1 and 0 with associated eigenvectors As expected (from Theorem 6.15(d) p. 371). these vectors are orthogonal. The corresponding orthonormal basis of eigenvectors i < r /2_ .Vv/57 Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices determines new axes x' and y' as in Figure 6.1. Hence if (2 - 1 391 / ' then 1 2 V 5 v'5 / P'AP 2 - I V 5V1 2 0 (i I'nder the transformation X / 'A ' or 2 , I , ' - 7 = * -7?2/ /-> v-> v - V + 4 ' ' -V 5 y 5 we have the new quadratic form (x') I (it//')2. Thus the original equation 2x2 Axy by2 36 may be written in the form (./')' M>(//)2 = 36 relative to a new coordinate system with the x'- and //'-axes in the directions of the first and second vectors ol 4. respectively. It is clear I hat this equation represents Figure 6.4 an ellipse. (See Figure 6.4.) Note thai the preceding matrix / ' has the form cos 9 sin 0 sin0 cos0 where 9 = cos'1 = ~ 2(i.(i . So / ' is the matrix representation of a rotal ion v 5 of R2 through the angle 0. Thus the change of variable X PX' can be ac-complished by this rotation of the X- and //-axes. There is another possibility 392 Chap. 6 Inner Product Spaces for P. however. II the eigenvector of A corresponding to the eigenvalue 6 is taken to be (I. 2) instead of ( 1.2). and the eigenvalues are interchanged, then we obtain the matrix / J _ _2_ - 2 1 V.75 7 5 . which is the matrix representation of a rotation through the angle 9 -9 VO 63.4. This possibility produces the same ellipse as the one in figure 0. I. but interchanges the names of the x'- and //'-axes. EXERCISES 1. Label the following statements as true or false. Assume that the under-lying inner product spaces are finite-dimensional. (a) Every unitary operator is normal. (b) Even- orthogonal operator is diagonalizable. (Q) A matrix is unitary if and only if it is invertible. (d) If two mat rices are unit arily equivalent. t hen t hey are also similar. (e) The sum ol unitary matrices is unitary. (f) The adjoint ol a unitary operator is unitary. (g) If T is an orthogonal operator on V. then [T j is an orthogonal matrix for any ordered basis / for V. (0) If all the eigenvalues of a linear operator are I. then the operator must be unitary or orthogonal. (i) A linear operator may preserve the norm, but not the inner prod-uct. 2. for each of the following matrices .1. find an orthogonal or unitary matrix / ' and a diagonal matrix D such thai P'AP = D. W . I 2 I) I) 3 - 3 / (C ) 3 - 3 , 5 /0 2 2 2 I I' (d) 2 (l 2 (e) 1 2 1 2 2 0 / VI I 2 3. Prove that the composite ol unitary orthogonal operators is unitary ort hogonalb Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 393 4. For z 6 C, define T~: C > C by T-;(394 Chap. 6 Inner Product Spaces 11. Find an orthogonal matrix whose first row is ( | , | , ^). 12. Let A be an n x n real symmetric or complex normal matrix. Prove that n (let 1,1. - ] ] A , . where the A,'s are the (not necessarily distinct) eigenvalues of A. 13. Suppose that .4 and 11 are diagonalizable matrices. Prove or disprove that A is similar to II if and only if A and 11 are unitarily equivalent. 14. Prove that if A and 11 are unitarily equivalent matrices, then A is pos-itive definite [semidefinite] if and only if II is positive definite semidef-inite]. (See t he definit ions in the exercises in Section (i. 1.) 15. Let U be a unitary operator on an inner product space V. and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) W: (b) W- is U-invariant. ('ontrast (b) with Exercise 16. 16. Find an example of a unitary operator U on an inner product space and a U-invariant subspace W such thai W - is not U-invariant. 17. Prove that a matrix that is both unitary and upper triangular must be a. diagonal mat rix. 18. Show that 'is unitarily equivalent to' is an equivalence relation on M n x n (C) . 19. Let W be a finite-dimensional subspace of an inner product space V. By Theorem 0.7 (p. 352) and the exercises of Section 1.3. V W :AN1 . Define U : V -V by U(e, I u2) = V - V2, where e, G W and v2 G W . Prove that U is a. self-adjoint unitary operator. 20. Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial isometry if there exists a subspace W of V such that |!U(.r)|| = ||;r|| for all x C W and U(.r) 0 for all x G W-. Observe that W need not be U-invariant. Suppose that U is such an operator and {v, r-> Uk} is an orthonormal basis for W. Prove the following results. (a) (U(:r).U(//)) = (x,y) for all x.y e W. Hint: Use Exercise 20 of Section 6.1. (b) {U(/']).U(r,) U(a-)} is an orthonormal basis for R(U). mm Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 395 (c) There exists an orthonormal basis 7 for V such that the first k columns of [U]7 form an orthonormal set and the remaining columns are zero. (d) Let {wi,iu-2, Wj} be an orthonormal basis for R(U)1- and 0 = {U(ci), U(e2) , . . . , J(vk), W],..., Wj}. Then 0 is an orthonormal basis for V. (e) Let T be the linear operator on V that satisfies T(U(t',;)) = v (1 < i < k) and T(v)i) = 0 (1 < i < 3). Then T is well defined, and T = U*. Hint: Show that (D(x),y) = (x,T(y)) for all x,y G 0. There are four cases. (f) U* is a partial isometry. This exercise is continued in Exercise 9 of Section 6.6. 21. Let A and B be n x n matrices that, are unitarily equivalent. (a) Prove that tr(AM) = ti(B*B). (b) Use (a) to prove that E I ^ I 2 = E I % I 2 -1,3-] t,j = ] (c) Use (b) to show- that the matrices 1 2 . fi 4 2 i) m , d ( l 1 are not unitarily equivalent. 22. Let V be a real inner product space. (a) Prove that any translation on V is a rigid motion. (b) Prove that the composite of any two rigid motions on V is a rigid motion on V. 23. Prove the following variation of Theorem 6.22: If / : V > V is a rigid motion on a finite-dimensional real inner product space V, then there exists a unique orthogonal operator T on V and a unique translation g on V such that f = T o g. 24. Let T and U be orthogonal operators on R2. Use Theorem 6.23 to prove the following results. (a) If T and U are both reflections about lines through the origin, then UT is a rotation. (b) If T is a rotation and U is a reflection about a line through the origin, then both UT and TU arc reflections about lines through the origin. 396 Chap. 6 Inner Product Spaces 25. Suppose that T and U are reflections of R2 about the respective lines L and L' through the origin and that 0 and v are the angles from the positive x-axis to L and L'. respectively. By Exercise 24. UT is a rotation. Find its angle of rotation. 26. Suppose that T and U are orthogonal operators on R such that T is the rotation by the angle o and U is the reflection about the line L through the origin. Let r be the angle from the positive .r-axis to L. By Exercise 24. both UT and TU are reflections about lines l_i and L2, respectively, through the origin. (a) bind the angle 9 from the positive rc-axis to L|. (b) bind the angle 0 from the positive .r-axis to L2. 27. Find new coordinates x',y' so that the following quadratic forms can be written as Xi(x') -t- A2(j/7) . (a) x2 + Axy + y2 (b) 2x2 + 2xy 4- 2r (c) x2 - Ylxy - Ay2 (d) 3./-2 + 2xy + 3//2 (e) x2 - 2.ry + y2 28. Consider the expression X AX, where X1 = (./. //. z) and .4 is as defined in Exercise 2(e). find a change of coordinates x'.y'.z' so that the preceding expression is of the form A| (.r')2 4- X->(y')2 + X%(z')2. 29. QR-Factorization. Let wi,W2,-.. ,w be linearly independent vectors in F'. and let L'i.r> rn be the orthogonal vectors obtained from W,W2 a',, by the Gram Schmidt process. Let Ui.uo ' be the orthonormal basis obtained by normalizing the r,'s. (a) Solving (1) in Section 6.2 for Wk in terms of u&, show that A--1 ifk = I VkUk + ^ (wk, Ui)U3 [(F) by ';ll i f j = * Rjk = { (wk,Uj) if j Ar. Prove A = QR. (c) Compute Q and R as in (b) for the 3 x 3 matrix whose columns are the vectors (1,1.0), (2,0,1). and (2,2.1). 9^ Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 397 (d) Since Q is unitary [orthogonal] and /? is upper triangular in (b), we have shown that every invertible matrix is the product of a uni-tary [orthogonal] matrix and an upper triangular matrix. Suppose that A G M n x n ( F ) is invertible and A = Q1Ri = Q>Il>- where Q-Qz Mx(/') are unitary and P.P, G Mx(/') are upper triangular. Prove that D = i?2/?j' is a unitary diagonal matrix. Hint: Use Exercise 17. (e) The QR factorization described in (b) provides an orthogonaliza-tion method for solving a linear system Ax b when A is in-vertible. Decompose A to QR. by the Gram Schmidt process or other means, where Q is unitary and />' is upper triangular. Then QRx b. and hence Rx = Q'b. This last system can be easily solved since P is upper triangular. Use the orthogonalization method and (c) to solve the system .r, 4- 2ar2 4- 2x3 - I Xj 2.r> - I I '2 + X3 = - 1 . 30. Suppose that 3 and *) are ordered bases for an //-dimensional real [com-plex] inner product space V. Prove thai if Q is an orthogonal [unitary] ii x ii matrix that changes 7-coordinates into ..^-coordinates, then (3 is orthonormal if and only i f ' is orthonormal. The following definition is used in Exercises 31 and 32. Definition. Let V be a finite-dimensional complex [real] inner product. sj)ce. and let u be a unit vector in V. Define the Householder operator H: V V by H ('./) = x - 2 (x, it) 11 for all x C V. 31. Let H be a Householder operator on a finite-dimensional inner product space V. Prove the following results. (a) H is linear. (b) H(.r) X if and only if X is orthogonal to u. (c) Hu(u) = -u. (d) H* = H and H2 = I, and hence H is a unitary [orthogonal] operator on V. (Note: If V is a real inner product space, then in the language of Sec-tion 6.11, H is a reflection.) 'At one time, because of its great stability, this method for solving large sys-tems of linear equations with a computer was being advocated as a better method than Gaussian elimination even though it requires about three times as much work. (Later, however. J. II. Wilkinson showed that if Gaussian elimination is done 'prop-erly,' then it is nearly as stable as the orthogonalization method.) 398 Chap. 6 Inner Product Spaces 32. Let V be a finite-dimensional inner product space over F. Let x and y be linearly independent vectors in V such that ||cc|| = y. (a) If F (', prove that there exists a. unit vector u in V and a complex number 0 with |0j 1 such that H(.;) 9y. Hint: Choose 9 so that (x.9y) is real, and set u r. r-Ax 9y). x 9y (b) If F = R. prove that there exists a unit vector // in V such thai ru(x) = y. 6.6 ORTHOGONAL PROJECTIONS AND THE SPECTRAL THEOREM In this sect ion. we rely heavily on Theorems 6.16 (p. 372) and 0.17 (p. 374) to develop an elegant representation of a normal (if /' C) or a self-adjoint (if F = R) operator T on a finite-dimensional inner product space. We prove that T can be written in the form A| T[ 4- A2T2 + 1 A/,T/,. where A,. X> A/, are the distinct eigenvalues of T and T| . T_>,.... T/,. arc orthogonal project ions. We must first develop some results about these special projections. We assume that the reader is familiar with the results about direct sums developed at the end of Section 5.2. The special case where V is a direct sum of two subspaces is considered in the exercises of Section 1.3. Recall from the exercises of Section 2.1 that if V = W] W2, then a linear operator T on V is t he projection on W, along W2 if. whenever ./ X] 4-.r2. with x 1 G Wi and x-z C WL>. we have T(.r) = x. By Exercise 20 of Section 2.1. we have R(T)=W, = b e V : T(x) =x d N(T) = W->. So V = R(T) N(T). Thus there is no ambiguity if we refer to T as a 'projection on W|' or simply as a 'projection.' fn fact, it can be shown (see Exercise 17 of Section 2.3) that T is a. projection if and only if T T2. Because V - W, W2 = W, W:i does not imply that W2 - W;i. we see that Wi does not uniquely determine T. For an orthogonal projection T. however, T is uniquely determined by its range. Definition. Let V be an inner product space, und let T: V V be a projection. We say that T is an orthogonal projection if R(T) = N(T) and N(T)1 - R(T). Note that by Exercise 13(c) of Section 6.2, if V is finite-dimensional, we need only assume that one of the preceding conditions holds, for example, if R(T) - N(T). then R(T) - R(T)' ' = N(T) . Now assume that W is a. finite-dimensional subspace of an inner product space V. In the notation of Theorem 6.6 (p. 350), we can define a function Sec. 6.6 Orthogonal Projections and the Spectral Theorem 399 T: V V by T(/y) u. It is easy to show that T is an orthogonal projection on W. We can say even more there exists exactly one orthogonal projection on W. For if T and U are orthogonal projections on W, then R(T) W R(U). Hence N(T) = R(T)J - R(U)J- = N(U), and since every projection is uniquely determined by its range and null space, we have T U. We call T the orthogonal projection of V on W. To understand the geometric difference between an arbitrary projection on W and the orthogonal projection on W. let V = R2 and W = span{(l, 1)}. Define U and T as in Figure 6.5. where T(v) is the foot of a. perpendicular from v on the line y ./ and U(ai,a2) = (r 400 Chap. 6 Inner Product Spaces whose coefficients are the Fourier coefficients o f / relative to the orthonormal set S. For this result, let W = span({/,: j < //}), and let T be the orthogo-nal projection of H on W. The corollary to Theorem 6.6 (p. 350) tells us that the best approximation to / by a function in W is T ( / ) = (fjj)fj An algebraic characterization of orthogonal projections follows in the next theorem. Theorem 6.24. Let V be an inner product space, and let T be a linear operator on V. Then T is an orthogonal projection if and only if T has an adjoint T* and T2 T = T'. Proof. Suppose that T is an orthogonal projection. Since T2 = T because T is a projection, we need only show that T* exists and T = T*. Now V = R(T) N(T) and R(T)1 N(T). Let x.y G V. Then we can write x = X 4 x-2 and y = yx 4- y>. where X.y G R(T) and .r2.y2 G N(T). Hence and ('-T(//)) - (,ri 4.r2 . / / i) = ( l i ,y i ) + {f2-!l} - {'! !!)) (T(.r).//) = (x-i.yi 4- !)>} = (xi,yi) + (x1,y2) - Sec. 6.6 Orthogonal Projections and the Spectral Theorem 401 Let V be a finite-dimensional inner product space, W be a subspace of V, and T be the orthogonal projection of V on W. We may choose an orthonormal basis j3 = {v, v402 Chap. 6 Inner Product Spaces (e) For x V, write x = x + x2 - h xk, where Xi Sec. 6.6 Orthogonal Projections and the Spectral Theorem 403 Corollary 2. If F = C, then T is unitary if and only if T is normal and |A| = 1 for every eigenvalue A ofT. Proof. If T is unitary, then T is normal and every eigenvalue of T has absolute value 1 by Corollary 2 to Theorem 6.18 (p. 382). Let T = Ai Tx + A2T2 -I- f- XkTk be the spectral decomposition of T. If |A| = 1 for every eigenvalue A of T, then by (c) of the spectral theorem, TT* = (AiTi + A2T2 + + AfcTfcXAiTi + A2T2 = |A1|2T1 + |A2|2T2 + --- + |Afc|2Tfc = Tj + T2 + + Tfc A*Tfc) Hence T is unitary. 1 Corollary 3. If F = C and T is normal, then T is self-adjoint if and only if every eigenvalue of T is real. Proof. Let T = AiTi + A2T2 -f- h A^T*. be the spectral decomposition of T. Suppose that every eigenvalue of T is real. Then T* = AiTi + A2T2 + - + AfcTfc = A,Ti + A2T2 + + XkTk = T. The converse has been proved in the lemma to Theorem 6.17 (p. 374). Corollary 4. Let T he as in the spectral theorem with spectral decom-position T = AiTi + A2T2 + + AfcTfc. Then each Tj is a polynomial in T. Proof Choose a polynomial gj (1 < j < k) such that #?(Aj) = 6^. Then 9j(V = Pi(Ai)Ti + gj{X2)T2 + + gj{Xk)Tk 5jT + S2jT2 - h SkjTk = Tj. I EXERCISES 1. Label the following statements as true or false. Assume that the under-lying inner product spaces are finite-dimensional. (a) All projections are self-adjoint. (b) An orthogonal projection is uniquely determined by its range. (c) Every self-adjoint operator is a linear combination of orthogonal projections. 404 Chap. 6 Inner Product Spaces (d) If T is a projection on W, then T(x) is the vector in W that is closest to x. (e) Every orthogonal projection is a unitary operator. 2. Let V = R2, W = span({(l,2)}), and /? be the standard ordered basis for V. Compute [T]^, where T is the orthogonal projection of V on W. Do the same for V = R3 and W = span({(l, 0,1)}). 3. For each of the matrices A in Exercise 2 of Section 6.5: (1) Verify that LA possesses a spectral decomposition. (2) For each eigenvalue of LA, explicitly define the orthogonal projec-tion on the corresponding eigenspace. (3) Verify your results using the spectral theorem. 4. Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal projection of V on W, then I T is the orthogonal projection of V on W-1. 5. Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal projection, prove that ||T(a:)|| < ||rr|| for all x e V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x 6 V? (b) Suppose that T is a projection such that ||T(.x)|| < ||x|| for x G V. Prove that T is an orthogonal projection. 6. Let T be a normal operator on a finite-dimensional inner product space. Prove that if T is a projection, then T is also an orthogonal projection. 7. Let T be a normal operator on a finite-dimensional complex inner prod-uct space V. Use the spectral decomposition AiTi + A2T2 + h XkTk of T to prove the following results. (a) If g is a polynomial, then (b) (c) (d) (e) (f) SCO = >(Ai)T< z=i If T' = To for some n, then T = To-Let U be a linear operator on V. Then U commutes with T if and only if U commutes with each Tj. There exists a normal operator U on V such that U2 = T. T is invertible if and only if Aj ^ 0 for 1 < i < k. T is a projection if and only if every eigenvalue of T is 1 or 0. Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse (g) T = T* if and only if every Aj is an imaginary number. 405 8. Use Corollary 1 of the spectral theorem to show that if T is a normal operator on a complex finite-dimensional inner product space and U is a linear operator that commutes with T, then U commutes with T*. 9. Referring to Exercise 20 of Section 6.5, prove the following facts about a partial isometry U. (a) U*U is an orthogonal projection on W. (b) UU*U = U. 10. Simultaneous diagonalization. Let U and T be normal operators on a finite-dimensional complex inner product space V such that TU = UT. Prove that there exists an orthonormal basis for V consisting of vectors that are eigenvectors of both T and U. Hint: Use the hint of Exercise 14 of Section 6.4 along with Exercise 8. 11. Prove (c) of the spectral theorem. 6.7* THE SINGULAR VALUE DECOMPOSITION AND THE PSEUDOINVERSE In Section 6.4, we characterized normal operators on complex spaces and self-adjoint operators on real spaces in terms of orthonormal bases of eigenvectors and their corresponding eigenvalues (Theorems 6.16, p. 372, and 6.17, p. 374). In this section, we establish a comparable theorem whose scope is the entire class of linear transformations on both complex and real finite-dimensional inner product spacesthe singular value theorem for linear transformations (Theorem 6.26). There are similarities and differences among these theorems. All rely on the use of orthonormal bases and numerical invariants. However, because of its general scope, the singular value theorem is concerned with two (usually distinct) inner product spaces and with two (usually distinct) orthonormal bases. If the two spaces and the two bases are identical, then the transformation would, in fact, be a normal or self-adjoint operator. Another difference is that the numerical invariants in the singular value theorem, the singular values, are nonnegative, in contrast to their counterparts, the eigen-values, for which there is no such restriction. This property is necessary to guarantee the uniqueness of singular values. The singular value theorem encompasses both real and complex spaces. For brevity, in this section we use the terms unitary operator and unitary-matrix to include orthogonal operators and orthogonal matrices in the context of real spaces. Thus any operator T for which (T(x),T(y)) = {x,y), or any matrix A for which (Ax, Ay) = (x, y), for all x and y is called unitary for the purposes of this section. 406 Chap. 6 Inner Product Spaces In Exercise 15 of Section 6.3, the definition of the adjoint of an operator is extended to any linear transformation T: V W, where V and W arc finite-dimensional inner product spaces. By this exercise, the adjoint T* of T is a linear transformation from W to V and [T*]^ = ([T]!)*, where 3 and 7 are orthonormal bases for V and W, respectively. Furthermore, the linear operator T*T on V is positive semidefinite and rank(T*T) rank(T) by Exercise 18 of Section 6.4. With these facts in mind, we begin with the principal result. Theorem 6.26 (Singular Value Theorem for Linear Transforma-tions). Let V and W be finite-dimensional inner product spaces, and let T: V > W be a linear transformation of rank r. Then there exist orthonormal bases {v,v2,... ,vn} for V and {u,u2,... ,um} for W and positive scalars o~ > > or such that T{Vi) = crjUj if 1 < i < r 0 ifi > r. (4) Conversely, suppose that the preceding conditions are satisfied. Then for 1 < i < n, Vi is an eigenvector of T*T with corresponding eigenvalue of if 1 < i < r and 0 if i > r. Therefore the scalars o_,o2,... ,o~r arc uniquely determined by T. Proof. We first establish the existence of the bases and scalars. By Ex-ercises 18 of Section 6.4 and 15(d) of Section 6.3, T*T is a positive semidef-inite linear operator of rank r on V; hence there is an orthonormal basis {v, v2, , v} for V consisting of eigenvectors of T*T with corresponding eigenvalues Aj, where Ai > A2 > > Ar > 0, and Aj = 0 for i > r. For 1 < i < r, define Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 407 and hence {ui,u2,... ,ur} is orthonormal. By Theorem 6.7(a) (p. 352), this set extends to an orthonormal basis {u, u2,..., ur,..., um} for W. Clearly T(vi) = GiUi if 1 < i < r. If i > r, then T*T(^j) = 0, and so T(vi) = 0 by Exercise 15(d) of Section 6.3. To establish uniqueness, suppose that {v,v2,... ,vn}, {u,u2,...,um}, and o~ > a2 > > or > 0 satisfy the properties stated in the first part of the theorem. Then for 1 < i < m and 1 < j < n, (T*(ui),Vj) = (ui,T(vj)) ai if i j < r 0 otherwise, and hence for any 1 < i < m, o~iVi if i = j < r 0 otherwise. (5) So for i < r, T*T(vi) = T*(cTjWj) = r. Therefore each Vi is an eigenvector of T*T with corresponding eigenvalue of if i < r and 0 if i > r. I Definition. The unique scalars o,o2,...,or in Theorem 6.26 are called the singular values of T. If r is less than both m and n, then the term singular value is extended to include or+i = = ok = 0, where k is the minimum of m and n. Although the singular values of a linear transformation T are uniquely de-termined by T, the orthonormal bases given in the statement of Theorem 6.26 are not uniquely determined because there is more than one orthonormal basis of eigenvectors of T*T. In view of (5), the singular values of a linear transformation T: V * W and its adjoint T* are identical. Furthermore, the orthonormal bases for V and W given in Theorem 6.26 are simply reversed for T*. Example 1 Let P2(R) and Pi(-R) be the polynomial spaces with inner products defined by (f(x),g(x))= / f{t)g{t)dt. - l 408 Chap. 6 Inner Product Spaces Let T: P2(/?) P(P) be the linear transformation defined by T(f(x)) = f'{x). Find orthonormal bases li = {v, v2,v3] for P2(R) and 7 = {u, u2} for Pi(H) such that T(i',) = OjUi for i = 1.2 and T(?;3) = 0, where o > a2 > 0 are the nonzero singular values of T. To facilitate the computations, we translate this problem into the corre-sponding problem for a matrix representation of T. Caution is advised here because not any matrix representation will do. Since the adjoint is defined in terms of inner products, we must use a matrix representation constructed from orthonormal bases for P2(li) and P(R) to guarantee that the adjoint of the matrix representation of T is the same as the matrix representation of the adjoint of T. (See Exercise 15 of Section 6.3.) For this purpose, we use the results of Exercise 21(a) of Section 6.2 to obtain orthonormal bases for P2(7?) and P^ft), respectively. Let A = m ; = 0 /3 0 0 0 v7!^ Then which has eigenvalues (listed in descending order of size) Ai = 15, A2 = 3, and A3 = 0. These eigenvalues correspond, respectively, to the orthonormal eigenvectors e3 = (0,0,1), e2 = (0,1,0), and c, = (1,0,0) in R3. Translating everything into the context of T. P2(i?). and Pi(/?), let and V-A =.. v/2 ui = / g ( 3 ^ - l ) , Then (3 {^1,^2,^3} is an orthonormal basis for P2(R) consisting of eigen-vectors of T*T with corresponding eigenvalues Ai. A2. and A3. Now set ~i A7 = N/15 and a2 = [X2 v3, the nonzero singular values of T, and take Mi = T(i'i) = - x and u2 = T{v2) = -7=, o~ V 2 cr2 s/2 to obtain the required basis 7 = {ui,u2} for Pi(/?). Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 409 We can use singular values to describe how a figure is distorted by a linear transformation. This is illustrated in the next example. Example 2 Let T be an invertible linear operator on R2 and S'={a:GR2 : | |3; | | = l}, the unit circle in R2. We apply Theorem 6.26 to describe 5 ' = T(5). Since T is invertible, it has rank equal to 2 and hence has singular values o > 0. Let {vi,v2} and (3 = {ui,u2} be orthonormal bases for R2 so that T(i>i) = o~iUi and T(f2) = a2u2, as in Theorem 6.26. Then /3 determines a coordinate system, which we shall call the aj'^/'-coordinate system for R , where the x'-axis contains u and the ?/-axis contains u2. For any vector u R2, if u = xu + x'2u2, then [u]^ = ( , X; is the coordinate vector of u relative to 0. We characterize S' in terms of an equation relating x[ and x2. For any vector v = xi^i + x2v2 G R2, the equation u = T(v) means that u = T(a;1vi + x2v2) = xiT(t;i) -f- :r2T('410 Chap. 6 Inner Product Spaces The singular value theorem for linear transformations is useful in its ma-trix form because we can perform numerical computations on matrices. We begin with the definition of the singular values of a matrix. Definition. Let A be an m x n matrix. We define the singular values of A to be the singular values of the linear transformation LA Theorem 6.27 (Singular Value Decomposition Theorem for Ma-trices). Let A be an m x n matrix of rank r with the positive singular values o > o2 > > ay, and let E be the ra x n matrix defined by Sij Oi ifi = j < r 0 otherwise. Then there exists an m x ra unitary matrix U and a n n x n unitary matrix V such that A = UXV*. Proof. Let T = L^: Fre -> Fm . By Theorem 6.26, there exist orthonormal bases f3 {v, v2,..., vn} for Fn and 7 = {u,u2,...,um} for Fm such that T(vi) OiUi for 1 < i < r and T(Vi) = 0 for i > r. Let U be the ra x ra matrix whose j th column is Uj for all j , and let V be the nxn matrix whose jth. column is Vj for all j. Note that both U and V are unitary matrices. By Theorem 2.13(a) (p. 90), the j t h column of AV is AVJ = Ojiij. Observe that the jth column of E is CTJCJ, where ej is the jth standard vector of Fm. So by Theorem 2.13(a) and (b), the j th column of UT. is given by U{o-3e3) = o~3U{e3) = OjUj. It follows that AV and UT, are ra x n matrices whose corresponding columns are equal, and hence AV = UE. Therefore A = AW* = UEV*. I Definition. Let A be an ra x n matrix of rank r with positive singular values o > o2 > > or. A factorization A = U'EV* where U and V are unitary matrices and E is the m x n matrix defined as in Theorem 6.27 is called a singular value decomposition of A. In the proof of Theorem 6.27, the columns of V are the vectors in (3, and the columns of U are the vectors in 7. Furthermore, the nonzero singular values of A are the same as those of L^; hence they are the square roots of the nonzero eigenvalues of A*A or of AA*. (See Exercise 9.) Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse Example 3 We find a singular value decomposition for A = First observe that for 411 1 - 1 1 - 1 >' = 1 7 3 * - i - j and U3 = _1_ 7e the set 0 = {^1,^2,^3} is an orthonormal basis for R3 consisting of eigen-vectors of A* A with corresponding eigenvalues Ai = 6, and A2 = A3 = 0. Consequently, o = /6 is the only nonzero singular value of A. Hence, as in the proof of Theorem 6.27, we let V be the matrix whose columns are the vectors in 0. Then E = y/Q 0 0 0 0 0 and V = / 1 J_ 1 V^ /2 V^ 1 -1 1 412 Chap. 6 Inner Product Spaces Furthermore, if A is invertible, then the representation is unique. Proof. By Theorem 6.27, there exist unitary matrices U and V and a diagonal matrix E with nonnegative diagonal entries such that A = t/EV*. So A = UYV* = UV*VY,V* = WP, where W = UV* and P = VEV*. Since W is the product of unitary matrices, W is unitary, and since E is positive semidefinite and P is unitarily equivalent to E, P is positive semidefinite by Exercise 14 of Section 6.5. Now suppose that A is invertible and factors as the products A = WP = ZQ. where W and Z are unitary and P and Q are positive semidefinite. Since A is invertible, it follows that P and Q are positive definite and invertible, and therefore Z*W = QP~X. Thus QP~X is unitary, and so / = (QF-'YiQp-1) = P-1Q2P~]. Hence P2 = Q2. Since both P and Q are positive definite', it follows that P = Q by Exercise 17 of Section 6.4. Therefore W = Z, and consequently the factorization is unique. I The factorization of a square matrix A as WP where W is unitary and P is positive semidefinite, is called a polar decomposit ion of A. Example 4 To find the polar decomposition of A = 11 - 5 - 2 10 , we begin by finding a sin-gular value decomposition WEV* of A. The object is to find an orthonormal basis 0 for R2 consisting of eigenvectors of A* A. It can be shown that * = 7 1 ( - 1 '2 = 7*2 ( I arc orthonormal eigenvectors of A* A with corresponding eigenvalues Ai 200 and A2 = 50. So 0 {v,v2} is an appropriate basis. Thus o = v200 = 10v2 and o2 = VoO = 5/2 are the singular values of A. So we have V = E = / l 0 / 2 0 5V2 Next, we find the columns u and u2 of U: Mi = Av = - I ) and Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse Thus U = Therefore, in the notation of Theorem 6.28, we have 413 W = UV* = and P = VZV* = The Pseudoinverse Let V and W be finite-dimensional inner product spaces over the same field, and let T: V W be a linear transformation. It is desirable to have a linear transformation from W to V that captures some of the essence of an inverse of T even if T is not invertible. A simple approach to this problem is to focus on the 'part' of T that is invertible, namely, the restriction of T to N(T)'1. Let L: N(T)1- R(T) be the linear transformation defined by L(x) = T(x) for all x G N(T)-1-. Then L is invertible, and we can use the inverse of L to construct a linear transformation from W to V that salvages some of the benefits of an inverse of T. Definition. Let V and W be finite-dimensional inner product spaces over the same field, and let T: V * W be a linear transformation. Let L: N(T)-1 R(T) be the linear transformation defined by L(x) = T(x) for all x G N(T)-1. Tlie pseudoinverse (or Moore-Penrose generalized inverse) of T, denoted by T', is defined as the unique linear transformation from W to V such that Tf(M) = L-HM) foryeR(T) 0 foryeRiT)2-. The pseudoinverse of a linear transformation T on a finite-dimensional inner product space exists even if T is not invertible. Furthermore, if T is invertible, then T* = T _ 1 because N(T)-1 = V, and L (as just defined) coincides with T. As an extreme example, consider the zero transformation To: V > W between two finite-dimensional inner product spaces V and W. Then R(To) = {0}, and therefore T* is the zero transformation from W to V. 414 Chap. 6 Inner Product Spaces We can use the singular value theorem to describe the pseudoinverse of a linear transformation. Suppose that V and W are finite-dimensional vector spaces and T: V > W is a linear transformation or rank r. Let {v,v2,..., vn} and {u,u2,... ,um} be orthonormal bases for V and W, respectively, and let oi > o2 > > or be the nonzero singular values of T satisfying (4) in Theo-rem 6.26. Then {v,v2,..., ty} is a basis for N(T)-1-, {vr+i,vr+2,..., vn} is a basis for N(T), {u,u2,... ,ur] is a basis for R(T), and {ur+,ur+2,... ,um} is a basis for R(T)X. Let L be the restriction of T to N(T)-1, as in the definition of pseudoinverse. Then L_1(UJ) Vi for 1 < i < r. Therefore Oi I Vi if 1 < i < r T f(Wj) = I Oi 0 if r < i < ra. (6) Example 5 Let T: P2(R) * Pi(-R) be the linear transformation defined by T(f(x)) f'{x), as in Example 1. Let 0 = {^1,^,^3} and 7 = {ui,u2} be the or-thonormal bases for P2(R) and P(R) in Example 1. Then a = / l5 and a 2 = v 3 are the nonzero singular values of T. It follows that T t ( y | , ) = T t ( W i ) = i V l ^ ^ 1), and hence Tt(*) = i ( 3 z 2 - 1 ) . Similarly, T*(l) = x. Thus, for any polynomial a + bx G P(R), T f(a + bx) = aV(l) + bV{x) = ax + ^{3x2 - 1). The Pseudoinverse of a Matrix Let A be an ra x n matrix. Then there exists a unique n x ra matrix B such that (Lyi)t: FTO > Fn is equal to the left-multiplication transformation LB- We call B the pseudoinverse of A and denote it by B A^. Thus (U) + = LA t . Let A be an m x n matrix of rank r. The pseudoinverse of A can be computed with the aid of a singular value decomposition A = UY.V*. Let 0 and 7 be the ordered bases whose vectors are the columns of V and U, Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 415 respectively, and let o > o2 > > ay be the nonzero singular values of A. Then 0 and 7 are orthonormal bases for F' and Fm, respectively, and (4) and (6) are satisfied for T = L^. Reversing the roles of 0 and 7 in the proof of Theorem 6.27, we obtain the following result. Theorem 6.29. Let A be anmxn matrix of rank r with a singular value decomposition A = UT.V* and nonzero singular values o > o2 > > ay. Let E* be the n x ra matrix defined by , I if i = j < r 0 otherwise. Then A^ VT^U*, and this is a singular value decomposition of A^. Notice that E^ as defined in Theorem 6.29 is actually the pseudoinverse of E. Example 6 We find A^ for the matrix A Since A is the matrix of Example 3, we can use the singular value decom-position obtained in that example: A = = UZV* = / 1 f 7 2 1 V2 -1 v/2 ' v ^ 0 0 0 0 0 fa 1 73 & 1 v/2 -1 V2 0 V6) 1 _2_ , v/6/ By Theorem 6.29, we have A* = VtfU* = & 1 V3 -1 1 V2 -1 V2 0 3s) 1 V _2_ 7e/ fa v 416 Chap. 6 Inner Product Spaces system has a unique solution for every /> G Fm if and only if A is invertible. in which case the solution is given by A~xb. Furthermore, if A is invertible. then A~l A*, and so the solution can be written as x = A^b. If, on the other hand, A is not invertible or the system Ax = bis inconsistent, then A^b still exists. We therefore pose the following question: In general, how is the vector ,4*6 related to the system of linear equations Ax = 6? In order to answer this question, we need the following lemma. Lemma. Let V and W be finite-dimensional inner product .spaces, and let T: V '-> W be linear. Then (a) T 'T is the orthogonal projection of V on N(T)'1. (b) TT' is the orthogonal projection of W on R(T). Proof. As in the earlier discussion, we define L: N(T)-1- > W by L(x) T{x) for all x G N(T) ' . If x G N(T)X, then T+T(a;) = L'1L(a:) = x, and if x G N(T), then TtT(./;) = T+(0) = 0. Consequently VT is the orthogonal projection of V on N(T)X. This proves (a). The proof of (b) is similar and is left as an exercise. I Theorem 6.30. Consider the system of linear equations Ax = b. where A is an in x // matrix and b G F'. If z = A^b. then z has the following properties. (a) If Ax = b is consistent, then z is the unique solution to the system having minimum norm. That is, z is a solution to the system, and if y is any solution to the system, then z < y with equality if and only if z = y. (b) If Ax = b is inconsistent, then z is the unique best approximation to a solution having minimum norm. That is, Az b < Ay b for any i) G F'. with equality if and only if Az = Ay. Furthermore, if Az = Ay, then ||z|| < || w|| with equality if and only if z y. Proof. For convenience, let T = L... (a) Suppose that Ax = b is consistent, and let z = A^b. Observe that b G R(T), and therefore Az = AA*b = TTf(/;) = 6 by part (b) of the lemma. Thus z is a solution to the system. Now suppose that y is any solution to the system. Then T+T(w) = A]Ay = AU) = z, and hence z is the orthogonal projection of y on N(T)'1' by part (a) of the lemma. Therefore, by the corollary to Theorem 6.6 (p. 350), we have that ll^ ll 5= ||M|| with equality if and only if z = y. (b) Suppose that Ax = b is inconsistent. By the lemma. Az = AA^b = TT'(6) = b is the orthogonal projection of b on R(T); therefore, by the corol-lary to Theorem 6.6 (p. 350), Az is the vector in R(T) nearest b. That is, if i Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 417 Ay is any other vector in R(T), then Az b < Ay b with equality if and only if Az = Ay. Finally, suppose that y is any vector in Fn such that Az Ay = c. Then A*c = A^Az = A^AA^b = A^b = z by Exercise 23; hence we may apply part (a) of this theorem to the system Ax = c to conclude that ||z|| < ||y|| with equality if and only if z = y. ii Note that the vector z = A'b in Theorem 6.30 is the vector x'o described in Theorem 6.12 that arises in the least squares application on pages 360 364. Example 7 Consider the linear systems X -I- x2 - X3 = 1 x + x2 - X3 = 1 and Xi + X2 - X3 = 1 x + x2 - x3 = 2. The first system has infinitely many solutions. Let A = '1 coefficient matrix of the system, and let b = 1 1 - 1 1 1 - 1 By Example 6, the and therefore D - 1 -lj is the solution of minimal norm by Theorem 6.30(a). The second system is obviously inconsistent. Let b = though Thus, al-z = A^b = -6 1 1 - 1 is not a solution to the second system, it is the 'best approximation' to a solution having minimum norm, as described in Theorem 6.30(b). 418 Chap. 6 Inner Product Spaces 1. 2. EXERCISES Label the following statements as true or false. (a) The singular values of any linear operator on a finite-dimensional vector space are also eigenvalues of the operator. (b) The singular values of any matrix A are the eigenvalues of A* A. (c) For any matrix A and any scalar c, if a is a singular value of A, then |c|a is a singular value of cA. (d) The singular values of any linear operator are nonnegative. (e) If A is an eigenvalue of a self-adjoint matrix A, then A is a singular value of A. (f) For any rnxn matrix A and any b G Fn, the vector A*b is a solution to Ax = b. (g) The pseudoinverse of any linear operator exists even if the operator is not invertible. Let T: V W be a linear transformation of rank r, where V and W are finite-dimensional inner product spaces. In each of the following, find orthonormal bases {vi,v2,..., vn} for V and {u, u2, , um} for W, and the nonzero singular values ai > a 2 > > a r of T such that T(vi) = OiUi for 1 < i < r. (a) T: R2 + R3 defined by T{x,x2) = (x, X + x2, X x2) (b) T: P2(R) - Pi(R), where T(f(x)) = f'(x), and the inner prod-ucts are defined as in Example 1 (c) Let V = W = span({l, sinx, cosx}) with the inner product defined by (/, g) = C f(t)g(t) dt, and T is defined by T( / ) = f + 2 / (d) T: C2 -> C2 defined by T(z1,z2) = ((1 - i)z2, (1 + i)zx + z2) 3. Find a singular value decomposition for each of the following matrices. (a) (d) (b) (e) 1 0 1 1 0 - 1 1-M 1 1 i i (c) ( 0 / I 0 1 V i I 0 V l I l I 0 - 1 1 1 - 2 1 1 1 4. Find a polar decomposition for each of the following matrices. ( a ) 0 - i ) ( b> ( 1 j { 5. Find an explicit formula for each of the following expressions. M Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 419 (a) T+ (x, x2, X3), where T is the linear transformation of Exercise 2(a) (b) T+(a + bx + ex2), where T is the linear transformation of Exer-cise 2(b) (c) T+(a + 6 sin a: + coos re), where T is the linear transformation of Exercise 2(c) (d) T+ iz ,z2) , where T is the linear transformation of Exercise 2(d) 6. Use the results of Exercise 3 to find the pseudoinverse of each of the following matrices. 7. (a) (d) (b) (e) 1 0 1 0 1 -H 1 - i 1 1N (c) (f) / l 0 1 V i I 0 ly I I I I 0 - l I I - 2 1 1 1 For each of the given linear transformations T: V W, (i) Describe the subspace Zi of V such that T*T is the orthogonal projection of V on Z. (ii) Describe the subspace Z2 of W such that TT* is the orthogonal projection of W on Z2. (a) T is the linear transformation of Exercise 2(a) (b) T is the linear transformation of Exercise 2(b) (c) T is the linear transformation of Exercise 2(c) (d) T is the linear transformation of Exercise 2(d) 8. For each of the given systems of linear equations, (i) If the system is consistent, find the unique solution having mini-mum norm. (ii) If the system is inconsistent, find the 'best approximation to a solution' having minimum norm, as described in Theorem 6.30(b). (Use your answers to parts (a) and (f) of Exercise 6.) X + x2 = 1 X + x2 + xs -- x4 2 (a) x + x2 = 2 (b) Xi - 2x3 + x4 = -1 X - X2 = 0 X X2 + .T3 + 4 = 2 9. Let V and W be finite-dimensional inner product spaces over F, and sup-pose that {v,v2,... ,vn} and {u,u2,...,um} are orthonormal bases for V and W, respectively. Let T: V W is a linear transformation of rank r, and suppose that ai > a 2 > > a r > 0 arc such that T(Mj) = OjUi if 1 < i < r 0 if r < i. 420 Chap. 6 Inner Product Spaces (a) Prove that {u,u2,... ,um} is a set of eigenvectors of TT* with corresponding eigenvalues Ai, A 2 , . . . , Am, where A, = (b) Let A be an ra x n matrix with real or complex entries. Prove that the nonzero singular values of A are the positive square roots of the nonzero eigenvalues of AA*, including repetitions. (c) Prove that TT* and T*T have the same nonzero eigenvalues, in-cluding repetitions. (d) State and prove a result for matrices analogous to (c). 10. Use Exercise 8 of Section 2.5 to obtain another proof of Theorem 6.27, the singular value decomposition theorem for matrices. 11. This exercise relates the singular values of a well-behaved linear operator or matrix to its eigenvalues. (a) Let T be a normal linear operator on an ?i-dimensional inner prod-uct space with eigenvalues Ai, A 2 , . . . , An. Prove that the singular values of T are |Ai|, |A2 | , . . . , |An|. (b) State and prove a result for matrices analogous to (a). 12. Let A be a normal matrix with an orthonormal basis of eigenvectors 0 {v,v2,... ,vn} and corresponding eigenvalues Ai, A 2 , . . . , An. Let V be the n x n matrix whose columns are the vectors in 0. Prove that for each i there is a scalar 0j of absolute value 1 such that if U is the n x n matrix with 9iVt as column i and E is the diagonal matrix such that Ejj = |Aj| for each i, then UT,V* is a singular value decomposition of A 13. Prove that if A is a positive semidefinite matrix, then the singular values of A are the same as the eigenvalues of A. 14. Prove that if A is a positive definite matrix and A UEV* is a singular value decomposition of A, then U = V. 15. Let A be a square matrix with a polar decomposition A = WP. (a) Prove that A is normal if and only if WP2 = P2W. (b) Use (a) to prove that A is normal if and only if WP = PW. 16. Let A be a square matrix. Prove an alternate form of the polar de-composition for A: There exists a unitary matrix W and a positive semidefinite matrix P such that A = PW. Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 421 17. Let T and U be linear operators on R2 defined for all (xi,x2) G R2 by T(xi,x2) = (xi,0) and U(xi,x2) = (xi + x2 ,0). (a) Prove that (UT)* ^ T*U*. (b) Exhibit matrices A and B such that AB is defined, but (AB)^ ^ 18. Let 4^ be an ra x n matrix. Prove the following results. (a) For any ra x ra unitary matrix G, (GAY = A^G*. (b) For any n x n unitary matrix H, (A//)* = ifM*. 19. Let A be a matrix with real or complex entries. Prove the following results. (a) The nonzero singular values of A are the same as the nonzero singular values of A*, which are the same as the nonzero singular values of A1. (b) (.4*)* = (A)*. (c) (At)-(A*)t . 20. Let A be a square matrix such that A2 = O. Prove that (A*)2 = O. 21. Let V and W be finite-dimensional inner product spaces, and let T: V > W be linear. Prove the following results. (a) TT*T = T. (b) T*TT*=T*. (c) Both T*T and TT* are self-adjoint. The preceding three statements are called the Penrose conditions, and they characterize the pseudoinverse of a linear transformation as shown in Exercise 22. 22. Let V and W be finite-dimensional inner product spaces. Let T: V > W and U: W > V be linear transformations such that TUT = T, UTU = U, and both UT and TU are self-adjoint. Prove that U = T*. 23. State and prove a result for matrices that is analogous to the result of Exercise 21. 24. State and prove a result for matrices that is analogous to the result of Exercise 22. 25. Let V and W be finite-dimensional inner product spaces, and let T: V * W be linear. Prove the following results. (a) If T is one-to-one, then T*T is invertible and T* = (T*T)~1T*. (b) If T is onto, then TT* is invertible and T* = T*(TT*)_1. 422 Chap. 6 Inner Product Spaces 26. Let V and W be finite-dimensional inner product spaces with orthonor-mal bases 0 and 7, respectively, and let T: V W be linear. Prove that ([T|j)t = [Tt]?. 27. Let V and W be finite-dimensional inner product spaces, and let T: V > W be a linear transformation. Prove part (b) of the lemma to Theorem 6.30: TT* is the orthogonal projection of W on R(T). 6.8* BILINEAR AND QUADRATIC FORMS There is a certain class of scalar-valued functions of two variables defined on a vector space that arises in the study of such diverse subjects as geometry and multivariable calculus. This is the class of bilinear forms. We study the basic properties of this class with a special emphasis on symmetric bilinear forms, and we consider some of its applications to quadratic surfaces and multivariable calculus. Bilinear Forms Definition. Let V be a vector space over a field F. A function II from the set V x V of ordered pairs of vectors to F is called a bilinear form on V if H is linear in each variable when the other variable is held fixed; that is, H is a bilinear form on V if (a) H{ax + x2, y) = aH(x], y) + H(x2, y) for a l lx , x 2 , y G V and a G F (b) H(x, mj 4- y2) aH(x, yi) 4 H(x, y2) for all x, y,y2 G V and a G F. We denote the set of all bilinear forms on V by #(V). Observe that an inner product on a vector space is a bilinear form if the underlying field is real, but not. if the underlying field is complex. Example 1 Define a function H: R2 x R2 H R by = 2ai&i 4- 3ai&2 + 4a2&i a2b2 for GR2. We could verify directly that II is a bilinear form on R2. However, it is more enlightening and less tedious to observe that if A = 2 4 ? v - w / , a2 and y then H(x,y) = x 'Ay. The bilinearity of H now follows directly from the distributive property of matrix multiplication over matrix addition. Sec. 6.8 Bilinear and Quadratic Forms 423 The preceding bilinear form is a special case of the next example. Example 2 Let V = Fn, where the vectors are considered as column vectors. For any A G Mn X n(F), define H: V x V - F by H(x, y) = xlAy for x, y G V. Notice that since x and y are n x 1 matrices and A is an nxn matrix, H(x,y) is a 1 x 1 matrix. We identify this matrix with its single entry. The bilinearity of H follows as in Example 1. For example, for a G F and x,x2,y G V, we have H(axi 4- x2,y) = (axi + x2)tAy = {ax 4 x)Ay = axAy 4 xAy = aH{x1,y) + H(x2,y). We list several properties possessed by all bilinear forms. Their proofs are left to the reader (see Exercise 2). For any bilinear form i J o n a vector space V over a field F, the following-properties hold. 1. If, for any x G V, the functions Lx, Rx: V * F are defined by Lx(y) = H{x,y) and Rx(y) = H(y,x) forallyGV, then Lx and R^ are linear. 2. H(0, x) = H(x, 0) = 0 for all x G V. 3. For all x, y,z,w G V, H(x + y,z + w) = H(x, z) 4 H(x, w) + H(y, z) 4 H(y, w). 4. If .7: V x V > F is defined by J(x,y) = H(y,x), then J is a bilinear form. Definitions. Let V be a vector space, let H and H2 be bilinear forms on V, and let a be a scalar. We define the sum Hi 4 H2 and the scalar product aH by the equations (f/i 4- H2){x,y) = Hx{x,y) 4 H2{x,y) and (aHi)(x,y) = a(Hi(x,y)) forallx,y G V. The following theorem is an immediate consequence of the definitions. 424 Chap. 6 Inner Product Spaces Theorem 6.31. For any vector space V, the sum of two bilinear forms and the product of a scalar and a bilinear form on V are again bilinear forms on V. Furthermore, B(V) is a vector space with respect to these operations. Proof. Exercise. | Let 0 = {vx,v2,... ,vn} be an ordered basis for an n-dimensional vector space V, and let H G (V). We can associate with // an ;/ x n matrix A whose entry in row i and column j is defined by Aij = H{vi, Vj) for i, j = 1 ,2 , . . . , n. Definition. The matrix A above is called the matrix representation of H with respect to the ordered basis 0 and is denoted bytppffi). We can therefore regard ijjp as a mapping from S(V) to Mxr ,(F). where F is the field of scalars for V, that takes a bilinear form H into its matrix representation ipp(H). We first consider an example and then show that ipp is an isomorphism. Example 3 Consider the bilinear form H of Example 1, and let r Then and So 0 = I3U = H BV2 = H B2 = H B22 =a H - 1 and B = $p(H). = 2 + 3 + 4 - 1 = 8, = 2 - 3 + 4 4 1 = 4 , = 2 4 3 - 4 + 1 = 2. = 2 - 3 - 4 - 1 = - 6 . MH) = 4 ,2 If 7 is the standard ordered basis for R , the reader can verify that '2 3N iMH) = 4 - 1 Sec. 6.8 Bilinear and Quadratic Forms 425 Theorem 6.32. For any n-dimcnsional vector space V over F and any ordered basis 0 for V, iftp: i?(V) > M r t X n(F) is an isomorphism. Proof. We leave the proof that ipp is linear to the reader. To show that ipp is one-to-one, suppose that ij^p(H) = O for some H G B(V). Fix Vi G 0, and recall the mapping LVi: V * F, which is linear by property 1 on page 423. By hypothesis, LVi(vj) = H(vi,Vj) = 0 for all Vj G 0. Hence LVi is the zero transformation from V to F. So H{vi, x) = LVi (x) = 0 for all x G V and v{ G 0. (7) Next fix an arbitrary y G V, and recall the linear mapping Ry: V > F defined in property 1 on page 423. By (7), Ry(vi) = H(vi,y) = 0 for all Vi G 0, and hence Ry is the zero transformation. So H{x,y) Ry(x) = 0 for all x,y G V. Thus H is the zero bilinear form, and therefore V/3 is one-to-one. To show that i/jp is onto, consider any A G M n X n (F) . Recall the isomor-phism p: V > Fn defined in Section 2.4. For x G V, we view (f)p(x) G Fn as a column vector. Let H: V X V > F be the mapping defined by H(x, y) = lMx)]*A^p(y)] for all x, y G V. A slight embellishment of the method of Example 2 can be used to prove that H G B{V). We show that tj>p(H) = A. Let viyv3 G 0. Then (i{vj) = e,j hence, for any i and j, H(Vi,Vj) = [Mvi)]tA[Mvj)} = eAe3- = Ai3. We conclude that tfipiH) = A and V^/3 is onto. | Corollary 1. For any n-dimensional vector space V, #(V) lias dimen-sion n2. Proof. Exercise. I The following corollary is easily established by reviewing the proof of Theorem 6.32. Corollary 2. Let V be an n-dimensional vector space over F with ordered basis 0. If H G B{1) and A G MXn(-F), ^cn ij)0{H) = A if and only ifH(x,y) = [^ (x)]*A[e(y)] for all x,y G V. The following result is now an immediate consequence of Corollary 2. Corollary 3. Let F be a field, n a positive integer, and 0 be the standard ordered basis for Fn. Then for any H G B(fn), there exists a unique matrix A G M,xn(F), namely, A = ipp{H), such that H(x, y) = xlAy for all x, y G F'. 426 Chap. 6 Inner Product Spaces Example 4 Define a function H: R2 x R2 -^ R by / / ' l h . n u r ' n : B - ^ - ^ ta&MS> 6i It can be shown that H is a bilinear form. We find the matrix A in Corollary 3 such that H{x,y) - x'Ay for all x,y G R2. Since Ai3 H(ci,e.j) for all i and j , we have A12 = d e t ( j f ) = l , Av = d e t | * ( ) ) '' {Sec. 6.8 Bilinear and Quadratic Forms 427 Suppose that A ipp(H) and B = ipy(H). Then for 1 < i, j < n, = ^,QkiVk and w3 =^TQrjVr. k= r=l Thus Bij = H{w.i,Wj) = H I 'Y^QkiVk.Wj J n = ^QkiH(vk,Wj) fc=sl = YL kiH I Vk> 5 Z QriVr) fc=l V 7-1 / Tt n = 22 ^kl S Q'jH(vk, Vr) fc=l r=l n n fc=l 7=1 71 fc=l r=l 7i fc=l TJ = yQlk(AQhJ = (QtAQ)lj. fe=i Hence B = Q*AQ. I The following result is the converse of Theorem 6.33. Corollary. Let V he an n-dimensional vector space with ordered basis 0, and let H be a bilinear form on V. For any n xn matrix B, if B is congruent to ipp(H), then there exists an ordered basis 7 for V such that ip^{H) = B. Furthermore, if B = Qt'iftp(H)Q for some invertible matrix Q, then Q changes 7-coordinates into 0-coordinates. Proof. Suppose that B = Q'ij)ft(H)Q for some invertible matrix Q and that 0 = {v,v2,... ,vn}. Let 7 = {w,w2,...,wn], where 2_, Qijvi fr 1 < i < M. 7=1 428 Chap. 6 Inner Product Spaces Since Q is invertible. 7 is an ordered basis for V, and Q is the change of coordinate matrix that changes 7-coordinates into /^-coordinates. Therefore, by Theorem 6.32, B = QtMH)Q=Symmetric Bilinear Forms Like the diagonalization problem for linear operators, there is an analogous diagonalization problem for bilinear forms, namely, the problem of determin-ing those bilinear forms for which there are diagonal matrix representations. As we will see, there is a close relationship between diagonalizable bilinear forms and those that are called symmetric. Definition. A bilinear form H on a vector space V is symmetric if H(x,y) = H(y,x) for all x.y G V. As the name suggests, symmetric bilinear forms correspond to symmetric matrices. Theorem 6.34. Let H be a bilinear form on a finite-dimensional vector space V, and let ii be an ordered basis for V. Then II is symmetric if and only ifif)p(H) is symmetric. Proof. Let 6 = {wi.va,.... v } and B = ip0{H). First assume that H is symmetric. Then for 1 < i,j < n, Bij = H(vi,vj) = H(v3yvi) = Bji, and it follows that B is symmetric. Conversely, suppose that B is symmetric. Let ./: V x V F, where F is the field of scalars for V, be the mapping defined by J(x,y) = H(y,x) for all x, y G V. By property 4 on page 423, ,/ is a bilinear form. Let C ipe{J)-Then, for 1 < i, j < n, CIJ=J(v1,vj) = H(vJ.vl) = BJI = Bu. Thus C = B. Since ijjp is one-to-one. we have ./ = / / . Hence H(y,x) = J(x,y) = H(.r.y) for all x.y G V, and therefore H is symmetric. LI Definition. A bilinear form H on a finite-dimensional vector space V is called diagonalizable if there is an ordered basis 0 for V such that 0^(H) is a diagonal matrix. Corollary. Let H be a diagonalizable bilinear form ou a finite-dimensional vector space V. Then H is symmetric. Sec. 6.8 Bilinear and Quadratic Forms 429 Proof Suppose that H is diagonalizable. Then there is an ordered basis 0 for V such that ij)@ (H) = D is a diagonal matrix. Trivially, D is a symmetric matrix, and hence, by Theorem 6.34, H is symmetric. 1 Unfortunately, the converse is not true, as is illustrated by the following example. Example 5 Let F = Z2,/ = F2, and H: V x V H F be the bilinear form defined by = ab2 +a2bx. Clearly H is symmetric. In fact, if 0 is the standard ordered basis for V, then A = # ( / / ) = ^ J V a symmetric matrix. We show that H is not diagonalizable. By way of contradiction, suppose that H is diagonalizable. Then there is an ordered basis 7 for V such that B 0 7 ( i / ) is a diagonal matrix. So by Theorem 6.33, there exists an invertible matrix Q such that B = Q* AQ. Since Q is invertible, it follows that rank (73) = rank(^4) = 2, and consequently the diagonal entries of B are nonzero. Since the only nonzero scalar of F is 1, Suppose that B = Q -1 0 0 1 Then = B = Q'AQ a c b d 0 1 1 0 a b c d ac + ac be + ad be + ad bd + bd But p + p = 0 for all p G F; hence ac + ac 0. Thus, comparing the row 1, column 1 entries of the matrices in the equation above, we conclude that 1 = 0, a contradiction. Therefore H is not diagonalizable. The bilinear form of Example 5 is an anomaly. Its failure to be diagonal-izable is due to the fact that the scalar field Z2 is of characteristic two. Recall 430 Chap. 6 Inner Product Spaces from Appendix C that a field F is of characterist ic two if 1 + 1 = 0 in F, If F is not of characteristic two, then 1 + 1 = 2 has a multiplicative inverse, which we denote by 1/2. Before proving the converse of the corollary to Theorem 6.34 for scalar fields that are not, of characteristic two, we establish the following Lemma. Lemma. Let H be a nonzero symmetric bilinear form on a vector space V over a field F not of characteristic two. Then there is a vector x in V such that H(x, x) ^ 0. Proof. Since H is nonzero, we can choose vectors u, v G V such that H(u,v) y 0. If H(u,u) ^ 0 or H(v,v) ^ 0, there is nothing to prove. Otherwise, set x = u + v. Then H(x,x) = H(u,u) + H{u,v) + H(v.u) + H{v,v) = 2H(u.v) ^ 0 because 2 ^ 0 and H{u. v) ^ 0. | Theorem 6.35. Let V be a finite-dimensional vector space over a field F not of characteristic two. Then every symmetric bilinear form on V is diagonalizable. Proof. We use mathematical induction on n = dim(V). If n 1, then every element of 1. and suppose that diin(V) = n. If // is the zero bilinear form on V. then trivially H is diagonalizable; so suppose that 77 is a nonzero symmetric bilinear form on V. By the lemma, there exists a nonzero vector x in V such that 77(x,x) / 0. Recall the function L,:: V F defined by L,(/y) = H(x,y) for all y G V. By property 1 on page 423, L^ is linear. Furthermore, since La(x) = II(x,x) ^ 0, Lx is nonzero. Consequently, rank(L,) = 1. and hence dim(N(LJ)) = n - 1. The restriction of 77 to N(L.r) is obviously a symmetric bilinear form on a vector space of dimension // 1. Tims, by the induction hypothesis, there exists an ordered basis {vi,v2,... ,vn i} for INKL^ ) such that II(vi,Vj) = 0 for i / j (1 < i.j < n - 1). Set yn = x. Then vn Sec. 6.8 Bilinear and Quadratic Forms 431 Diagonalization of Symmetric Matrices Let A be a symmetric n x n matrix with entries from a field F not of characteristic two. By the corollary to Theorem 6.35, there are matrices Q,D G M n x n ( F ) such that Q is invertible, D is diagonal, and Ql AQ = D. We now give a method for computing Q and D. This method requires familiarity with elementary matrices and their properties, which the reader may wish to review in Section 3.1. If E is an elementary nxn matrix, then AE can be obtained by performing an elementary column operation on A. By Exercise 21, ElA can be obtained by performing the same operation on the rows of A rather than on its columns. Thus E'AE can be obtained from A by performing an elementary operation on the columns of A and then performing the same operation on the rows of AE. (Note that the order of the operations can be reversed because of the associative property of matrix multiplication.) Suppose that Q is an invertible matrix and D is a diagonal matrix such that QlAQ = D. By Corollary 3 to Theorem 3.6 (p. 159), Q is a product of elementary matrices, sayQ = E1E2---Ek. Thus D = Q'AQ = _i EAE,E2 Ek. From the preceding equation, we conclude that by means of several elemen-tary column operations and the corresponding row operations, A can be trans-formed into a diagonal matrix D. Furthermore, if E,E2,... ,Ek are the elem,entary matrices corresponding to these elementary column operations in-dexed in the order performed, and if Q EE2 - Ek, then Q'AQ = D. Example 6 Let A be the symmetric matrix in M:ix;i(R) defined by We use the procedure just described to find an invertible matrix Q and a diagonal matrix D such that Q'AQ = D. We begin by eliminating all of the nonzero entries in the first row and first column except for the entry in column 1 and row 1. To this end, we add the first column of A to the second column to produce a zero in row 1 and column 2. The elementary matrix that corresponds to this elementary column operation is E, = 432 Chap. 6 Inner Product Spaces We perforin the corresponding elementary operation on the rows of AE to obtain /1 0 3N EAEi = 0 1 4 | . 3 4 1 We now use the first column of E[AE to eliminate the 3 in row 1 column 3, and follow this operation with the corresponding row operation. The corre-sponding elementary matrix E2 and the result of the elementary operations E^EAEXE2 are, respectively, E,= 0 0 0 1 0 - 3 0 1 and EEAEiE2 = 1 0 0 1 0 4 0 4 - 8 Finally, we subtract 4 times the second column of E2EAEE2 from the third column and follow this with the corresponding row operation. The cor-responding elementary matrix E3 and the result of the elementary operations E!iE2EAEE2E:i are, respectively, E3 = 1 0 0 1 0 0 0 - 4 1 and E^El2EAElE2E3 = 0 0 0 1 0 0 0 -24 Since we have obtained a diagonal matrix, the process is complete. So we let / I 1 - 7 / I 0 0N Q = E1E2E3 = I 0 1 - 4 and D = I 0 1 0 0 0 1/ 0 0 -24, to obtain the desired diagonalization Q' AQ = D. The reader should justify the following method for computing Q without recording each elementary matrix separately. The method is inspired by the algorithm for computing the inverse of a matrix developed in Section 3.2. We use a sequence of elementary column operations and corresponding row operations to change the n x 2n matrix (AI) into the form (DB), where D is a diagonal matrix and B = Q*. It then follows that D = Q'AQ. Starting with the matrix A of the preceding example, this method pro-duces the following sequence of matrices: (AI)= -1 1 3 - 1 2 1 3 1 1 1 1 3 0 3 1 1 1 1 Sec. 6.8 Bilinear and Quadratic Forms 433 Therefore F> = Ql = (l 1 -V and Q = I 0 1 - 4 VO 0 1, Quadratic Forms Associated with symmetric bilinear forms are functions called quadratic forms. Definition. Let V be a vector space over F. A function K: V F is called a quadratic form if there exists a symmetric bilinear form 77 G #(V) such that K(x) = H(x, x) for all x G V. (8) If the field F is not of characteristic two, there is a one-to-one correspon-dence between symmetric bilinear forms and quadratic forms given by (8). In fact, if K is a quadratic form on a vector space V over a field F not of characteristic two, and K(x) = 77(x, x) for some symmetric bilinear form 77 on V, then we can recover 77 from K because H(x,y) = -[K(x + y) K{x) - K(y)} (9) (See Exercise 16.) Example 7 The classic example of a quadratic form is the homogeneous second-degree polynomial of several variables. Given the variables t,t2,...,tn that take values in a field F not of characteristic two and given (not necessarily distinct) scalars aij (1 < i < j < n), define the polynomial f(h,t2,..., tn) = ^2 a-ijtitj-i434 Chap. 6 Inner Product Spaces Any such polynomial is a quadratic form. In fact, if 3 is the standard or-dered basis for F', then the symmetric bilinear form 77 corresponding to the quadratic form / has the matrix representation ipn(H) = A. where A 7 A'* ~~ an if t = j ai3 if i^j. To see this, apply (9) to obtain H(ei,e3) = A j from the quadratic form K, and verify that / is computable from H by (8) using / in place of K. For example, given the polynomial / ( ' i . ' 2 . h) = 2/'f - t + 6tit2 - 4t2t3 with real coefficients, let Setting H(x,y) = x' Ay for all x.y G R' we see that % f(ti,t2)t3) = (tut2,t3)A ( t2 Quadratic Forms Over the Field R Since symmetric matrices over 7? are orthogonally diagonalizable (see The-orem 6.20 p. 384), the theory of symmetric bilinear forms and quadratic forms on finite-dimensional vector spaces over 77 is especially nice. The following theorem and its corollary are useful. Theorem 6.36. Let V be a finite-dimensional real inner product space, and let H be a symmetric bilinear form on V. Then there exists an orthonor-mal basis 0 for V such that /',*( 77) is a diagonal matrix. Proof. Choose any orthonormal basis 7 = {vi,v2,... ,vn} for V, and let A */>-v(T7). Since A is symmetric, there exists an orthogonal matrix Q and a diagonal matrix D such that D Q' AQ by Theorem 6.20. Let 0 = {wi,w2,..., Wn} be defined by ivj = YJ QijVi for 1 < 7 < n. 7=1 By Theorem 6.33, ^#(77) = D. Furthermore, since Q is orthogonal and 7 is orthonormal, 0 is orthonormal by Exercise 30 of Section 6.5. SI L . Sec. 6.8 Bilinear and Quadratic Forms 435 Corollary. Let K be a quadratic form on a finite-dimensional real inner product space V. There exists an orthonormal basis 0 = {v,v2,... ,vn} for V and scalars Ai, A 2 , . . . , An (not necessarily distinct) such that if x G V and n x = ^2siVi, Si G 7?, ;=i then K(x) = J2XiS2. In fact, if 77 is the symmetric bilinear form determined by K, then 0 can be chosen to be any orthonormal basis for V such that il>p(H) is a diagonal matrix. Proof. Let 77 be the symmetric bilinear form for which K{x) = H(x,x) for all x G V. By Theorem 6.36, there exists an orthonormal basis 0 {v,v2, ... ,vn} for V such that tpp(H) is the diagonal matrix /Ai 0 0 A2 D = 0 0 0 0 A n / Let x G V, and suppose that x = X^iLi SiVi- Then K(x)=H(x,x) = [p(x)]tD[(i(x)} = (s1,s2,...,sn)D fsi s2 = I > f . I 7=1 Example 8 For the homogeneous real polynomial of degree 2 defined by / ( t i , t 2 ) = 5i2 + 2r;2 + 4 l l,i2, (10) we find an orthonormal basis 7 = {xi ,x 2} for R2 and scalars Ai and A2 such that if 1 ) G R2 and ( '} ] = SiXi + s2x2 , then f{t,t2) = Xis2 + A2A,2>. We can think of s and $2 as the coordinates of (ti,t2) relative to 7. Thus the polynomial f(ti,t2), as an expression involving 436 Chap. 6 Inner Product Spaces the coordinates of a point with respect to the standard ordered basis for R2, is transformed into a new polynomial g(si,s2) = XiS2 + A2s2 interpreted as an expression involving the coordinates of a point relative to the new ordered basis 7. Let 77 denote the symmetric bilinear form corresponding to the quadratic form defined by (10), let 0 be the standard ordered basis for R2, and let i4 = ^ ( H ) . Then Next, we find an orthogonal matrix Q such that Q'AQ is a diagonal matrix. For this purpose, observe that Ai = 6 and A2 = 1 are the eigenvalues of A with corresponding orthonormal eigenvectors ', = 7 i ( i ) d W2 = 7 s ( - 2 ) ' Let 7 = {vi,v2}. Then 7 is an orthonormal basis for R2 consisting of eigen-vectors of A. Hence, setting Q J _ (2 1 we see that Q is an orthogonal matrix and '6 0 QlAQ = 0 1 Clearly Q is also a change of coordinate matrix. Consequently, '6 0N V7(77) = Q*il>p{H)Q = QlAQ = y (| L Thus by the corollary to Theorem 6.36, K(x) = 6sl + si for any x = sv 1 + s2i>2 G R2. So g(si, s2) = 6s2 + s2 . The next example illustrates how the theory of quadratic forms can be applied to the problem of describing quadratic surfaces in R3. Example 9 Let S be the surface in R3 defined by the equation , 2*f + 6*i*2 + 5*1 - 2*2*3 + 2*1 + 3*i - 2*2 - *3 + 14 = 0. (11) Sec. 6.8 Bilinear and Quadratic Forms 437 Then (11) describes the points of S in terms of their coordinates relative to 0, the standard ordered basis for R3. We find a new orthonormal basis 7 for R3 so that the equation describing the coordinates of 438 Chap. 6 Inner Product Spaces Figure 6.7 We are now ready to transform (11) into an equation involving coordinates relative to 7. Let x (ti, t2, t3) G R3, and suppose that x = S1V1 -r82v2+s$V3. Then, by Theorem 2.22 (p. I l l ) , and therefore 10 V35 Si 3,s2 3.S3 *i = 7= + *2 = 5.s2 2s3 /3o~ /l and *a = 3.S1 s2 S3 '35 VT4 Sec. 6.8 Bilinear and Quadratic Forms Thus 3*i 2*2 *3 = 439 14s? = -V14s 3 . Combining (11), (12), and the preceding equation, we conclude that if x G R3 and x = Sifi + S2^2 + S3M3, then x G S if and only if 2s^ + lsA2 - /Us3 + 14 = 0 or s3 = ^ s ? + & + Vu. Consequently, if we draw new axes x',y', and z' in the directions of V,v2, and V3, respectively, the graph of the equation, rewritten as z .'2 (xT + ^iy') 14, 7 v ' 2 coincides with the surface S. We recognize S to be an elliptic paraboloid. Figure 6.7 is a sketch of the surface S drawn so that the vectors V, v2 and V3 are oriented to lie in the principal directions. For practical purposes, the scale of the z' axis has been adjusted so that the figure fits the page. The Second Derivative Test for Functions of Several Variables We now consider an application of the theory of quadratic forms to mul-tivariable calculusthe derivation of the second derivative test for local ex-trema of a function of several variables. We assume an acquaintance with the calculus of functions of several variables to the extent of Taylor's theorem. The reader is undoubtedly familiar with the one-variable version of Taylor's theorem. For a statement and proof of the multivariable version, consult, for example, An Introduction to Analysis 2d ed, by William R. Wade (Prentice Hall, Upper Saddle River, N.J., 2000). Let z /(*i, *2, , *n) be a fixed real-valued function of n real variables for which all third-order partial derivatives exist and are continuous. The function / is said to have a local maximum at a point p G Rn if there exists a 6 > 0 such that f(p) > f(x) whenever x p|| < 8. Likewise, / has a local minimum at p G Rn if there exists a 5 > 0 such that f(p) < f(x) whenever ||x p < 6. If / has either a local minimum or a local maximum at p, we say that / has a local extremum at p. A point p G Rn is called a critical point of / if df(p)/dti = 0 for i = 1 ,2 , . . . , n. It is a well-known fact that if / has a local extremum at a point p G Rn, then p is a critical point of / . For, if / has a local extremum at p = (pi,p2, iPn)-, then for any i = 1 ,2 , . . . , n the 440 Chap. 6 Inner Product Spaces function i defined by 4>i(t) = f(pi,p2,... ,pi-,t,pi+i,... ,pn) has a local extremum at * = pi. So, by an elementary single-variable argument, df(p) d&ipi] Ok dt = 0. Thus p is a critical point of / . But critical points are not necessarily local extrema. The second-order partial derivatives of / at a critical point p can often be used to test for a local extremum at p. These partials determine a matrix A(p) in which the row i, column j entry is d2f{p) (dti)(dtj)-This matrix is called the Hessian matrix of / at p. Note that if the third-order partial derivatives of / are continuous, then the mixed second-order partials of / at p are independent of the order in which they are taken, and hence A(p) is a symmetric matrix. In this case, all of the eigenvalues of A(p) are real. Theorem 6.37 (The Second Derivative Test). Let / ( t i , * 2 , . . . , *n) be a real-valued function in n real variables for which all third-order partial derivatives exist and arc continuous. Let p = (p,p2,... ,pn) be a critical point of f, and let A(p) be the Hessian of f at p. (a) If all eigenvalues of A(p) arc positive, then f has a local minimum at p. (b) If all eigenvalues of A(p) are negative, then f has a local maximum at p. (c) 7f A(p) has at least one positive and at least one negative eigenvalue, then f has no local extremum at p (p is called a saddle-point of f). (d) If rank(A(p)) < n and A(p) does not have both positive and negative eigenvalues, then the second derivative test is inconclusive. Proof. If p ^ 0, we may define a function g: Rn 77 by 0(*i ,* 2 , . . . , t n) = /(*i +pl,t2 +p2,...,pn +*n) - f(p)-The following facts are easily verified. 1. The function / has a local maximum [minimum] at p if and only if g has a local maximum [minimum] at 0 (0 ,0 , . . . , 0). 2. The partial derivatives of g at 0 are equal to the corresponding partial derivatives of / at p. 3. 0 is a critical point of g. Sec. 6.8 Bilinear and Quadratic Forms 441 In view of these facts, we may assume without loss of generality that p = 0 and f(p) = 0. Now we apply Taylor's theorem to / to obtain the first-order approxima-tion of / around 0. We have /(*. ,, , . . - . * . ) - / W + E T T ^ + S E 7=1 d2f(0) dU ^ ' 2 ^(dti)(dtj) *z*j+5(*i,*2,. . . ,*) 1 n a2 /( f l ) 2 ^ ( ^ ) ( a * j *i*j + 5(*i,*2, ,*n)> where 5 is a real-valued function on Rn such that lim 5(x) lim 5(*i,*2, . ,*n) x-0 ||x[|2 (ti,ta,...,*n)-0 *i + *i + r- *n ? = 0. (13) (14) Let 7f: Rn 77 be the quadratic form defined by 1 y d2f(0) K (ti *2 v / 2 .^{dt^dt, 'Cjtj, (15) 77 be the symmetric bilinear form corresponding to K, and 0 be the standard ordered basis for Rn. It is easy to verify that ip@(H) $A(p). Since A(p) is symmetric, Theorem 6.20 (p. 384) implies that there exists an orthogonal matrix Q such that QtA{p)Q = /Ai 0 0 A2 0 0 0 0 XnJ is a diagonal matrix whose diagonal entries are the eigenvalues of A(p). Let 7 = {v,v2,... ,vn} be the orthogonal basis for Rn whose ith vector is the ith column of Q. Then Q is the change of coordinate matrix changing 7-coordinates into /^-coordinates, and by Theorem 6.33 ^7(77) = QlMH)Q = ^QtA(p)Q = ' * 0 0 0 T/ 442 Chap. 6 Inner Product Spaces Suppose that A(p) is not the zero matrix. Then A(p) has nonzero eigen-values. Choose e > 0 such that e < |Aj|/2 for all Aj ^ 0. By (14), there exists S > 0 such that for any x G Rn satisfying 0 < ||x|| < 6, we have |S(x)| < e||x||2. Consider any x G Rn such that 0 < ||x|| < S. Then, by (13) and (15), and hence f(x)-K(x) = S(x) SiVt. Then i= Z* 1 and K(x) = - y^Xn 2 7=1 Combining these equations with (16), we obtain 7=1 ^ ' i=l > ' Now suppose that all eigenvalues of A(p) are positive. Then ^A, - f > 0 for all i, and hence, by the left inequality in (17), / ( 0 ) = 0 < E ( i A , - e V s 2 < / ( x ) . 7=1 ^ ' Thus f(0) < / (x ) for ||x|| < 6, and so / has a local minimum at 0. By a similar argument using the right inequality in (17), we have that if all of the eigenvalues of A(p) are negative, then / has a local maximum at 0. This establishes (a) and (b) of the theorem. Next, suppose that A(p) has both a positive and a negative eigenvalue, say, Aj > 0 and Aj < 0 for some i and j. Then ^Aj - r > 0 and ^Xj + e < 0. Let s be any real number such that 0 < |s| < 6. Substituting x = svi and x = SVJ into the left inequality and the right inequality of (17), respectively, we obtain f(0) = 0Sec. 6.8 Bilinear and Quadratic Forms 443 To show that the second-derivative test is inconclusive under the condi-tions stated in (d), consider the functions /(*i,*2) = 4 - 4 and g{tiM) = tl+4 at p = 0. In both cases, the function has a critical point at p, and '2 0N A(p) = 0 0 However, / does not have a local extremum at 0, whereas g has a local minimum at 0. I Sylvester's Law of Inertia Any two matrix representations of a bilinear form have the same rank because rank is preserved under congruence. We can therefore define the rank of a bilinear form to be the rank of any of its matrix representations. If a matrix representation is a diagonal matrix, then the rank is equal to the number of nonzero diagonal entries of the matrix. We confine our analysis to symmetric bilinear forms on finite-dimensional real vector spaces. Each such form has a diagonal matrix representation in which the diagonal entries may be positive, negative, or zero. Although these entries are not unique, we show that the number of entries that are positive and the number that are negative are unique. That is, they are independent of the choice of diagonal representation. This result is called Sylvester's law of inertia. We prove the law and apply it to describe the equivalence classes of congruent symmetric real matrices. Theorem 6.38 (Sylvester's Law of Inertia). Let 77 be a symmetric bilinear form on a finite-dimensional real vector space V. Then the number of positive diagonal entries and the number of negative diagonal entries in any diagonal matrix representation of H are each independent of the diagonal representation. Proof. Suppose that 0 and 7 are ordered bases for V that determine di-agonal representations of 77. Without loss of generality, we may assume that 0 and 7 are ordered so that on each diagonal the entries are in the order (of positive, negative, and zero. It suffices to show that both representations have the same number of positive entries because the number of negative en-tries is equal to the difference between the rank and the number of positive entries. Let p and q be the number of positive diagonal entries in the matrix representations of 77 with respect to 0 and 7, respectively. We suppose that p q and arrive at a contradiction. Without loss of generality, assume that p < q. Let 0 = {v,v2,...,vp,...,vr,...,vn} and 7 = {wi,w2,... ,wq,... ,wr,... ,wn}, u 444 Chap. 6 Inner Product Spaces where r is the rank of 77 and n = dim(V). Let L: V * RP+r t be the mapping defined by L(x) = (H(x,vi), H(x,v2),..., H{x,vp), H{x,wq+i),..., H(x,wr)). It is easily verified that L is linear and rank(L) < p + r q. Hence nullity(L) > n (p + r q) > n r. So there exists a nonzero vector VQ such that vo ^ span({?;r+i, vr+2,..., vn}), but vo G N(L). Since v0 G N(L). it follows that 77(t'0,Vj) = 0 for i < p and H{vo,Wi) = 0 for q < i < r. Suppose that Wi. >o = ^a3v3 = E ' V -j=i j=i For any i < p, H(v0,Vi) = 77 I E a i v i J v * I = ^2ajH(v3,Vi) = a.iH{vi,Vi). j-i J i - i But for i < p, we have H{vi,Vj) > 0 and H(v{),Vi) = 0, so that Uj = 0. Similarly, 6j = 0 for q + 1 < i < r. Since VQ is not in the span of {vr+, t'r+2i > Mn}, it follows that at ^ 0 for some p < i < r. Thus In 7t Tt r H(vo,Vo)=Hl^a3v3,^aiVi=Yla>2jH{v3,v3)= E oJ/T( i , t; j)J^(M'J,M;J)>0. j=l 7=1 / j=l j=P+l So 77(fn,Mo) < 0 and H(vo,vo) > 0, which is a contradiction. We conclude that p q. 1 Definitions. The number of positive diagonal entries in a diagonal representation of a symmetric bilinear form on a real vector space is called the index of the form. The difference between the number of positive and the number of negative diagonal entries in a diagonal representation of a symmetric bilinear form is called the signature of the form. The three terms rank, index, and signature are called the invariants of the bilinear form because they are invariant with respect to matrix representations. These same terms apply to the associated quadratic form. Notice that the values of any two of these invariants determine the value of the third. Sec. 6.8 Bilinear and Quadratic Forms 445 Example 10 The bilinear form corresponding to the quadratic form K of Example 9 has a 3 x 3 diagonal matrix representation with diagonal entries of 2, 7, and 0. Therefore the rank, index, and signature of K are each 2. Example 11 The matrix representation of the bilinear form corresponding to the quadratic form K(x,y) = x2 y2 on R2 with respect to the standard ordered basis is the diagonal matrix with diagonal entries of 1 and 1. Therefore the rank of K is 2, the index of K is 1, and the signature of K is 0. Since the congruence relation is intimately associated with bilinear forms, we can apply Sylvester's law of inertia to study this relation on the set of real symmetric matrices. Let A be an n x n real symmetric matrix, and suppose that D and E are each diagonal matrices congruent to A By Corollary 3 to Theorem 6.32, A is the matrix representation of the bilinear form 77 on Rn defined by H(x,y) = xlAy with respect to the standard ordered basis for Rn. Therefore Sylvester's law of inertia tells us that D and E have the same number of positive and negative diagonal entries. We can state this result as the matrix version of Sylvester's law. Corollary 1 (Sylvester's Law of Inertia for Matrices). Let A be a real symmetric matrix. Then the number of positive diagonal entries and the number of negative diagonal entries in any diagonal matrix congruent to A is independent of the choice of the diagonal matrix. Definitions. Let A be a real symmetric matrix, and let D be a diagonal matrix that is congruent to A. The number of positive diagonal entries of D is called the index of A. The difference between the number of positive diagonal entries and the number of negative diagonal entries of D is called the signature of A. As before, the rank, index, and signature of a matrix are called the invariants of the matrix, and the values of any two of these invariants determine the value of the third. Any two of these invariants can be used to determine an equivalence class of congruent real symmetric matrices. Corollary 2. Two real symmetric n x n matrices are congruent if and only if they have the same invariants. Proof. If A and B are congruent n x n symmetric matrices, then they are both congruent to the same diagonal matrix, and it follows that they have the same invariants. Conversely, suppose that A and B are nx n symmetric matrices with the same invariants. Let D and E be diagonal matrices congruent to A and B, 446 Chap. 6 Inner Product Spaces respectively, chosen so that the diagonal entries are in the order of positive, negative, and zero. (Exercise 23 allows us to do this.) Since A and B have the same invariants, so do D and E. Let p and r denote the index and the rank, respectively, of both D and E. Let di denote the ith diagonal entry of D, and let Q be the n x n diagonal matrix whose ith diagonal entry qi is given by ( 1 -7= if 1 < i < p di Qi = S 1 if p < i < r Then QtDQ Jpr, where l-di 1 if r < i. O Jpr I *rp p o It follows that A is congruent to Jpr- Similarly, B is congruent to Jpr, and hence A is congruent to B. [ The matrix Jpr acts as a canonical form for the theory of real symmet-ric matrices. The next corollary, whose proof is contained in the proof of Corollary 2, describes the role of Jpr. Corollary 3. A real symmetric n x n matrix A has index p and rank r if and only if A is congruent to Jpr (as just defined). Example 12 Let A = 1 1 1 2 3 1 '3 1 ' 1 / 77 = I 2 1 1 2 3 2 l 2 1 and C = We apply Corollary 2 to determine which pairs of the matrices A, B, and C are congruent. The matrix A is the 3 x 3 matrix of Example 6, where it is shown that A is congruent to a diagonal matrix with diagonal entries 1, 1, and 24. Therefore, A has rank 3 and index 2. Using the methods of Example 6 (it is not necessary to compute Q), it can be shown that B and C are congruent, respectively, to the diagonal matrices and x- - Sec. 6.8 Bilinear and Quadratic Forms 447 It follows that both A and C have rank 3 and index 2, while B has rank 3 and index 1. We conclude that A and C are congruent but that B is congruent to neither A nor C. EXERCISES 1. Label the following statements as true or false. (a) Every quadratic form is a bilinear form. (b) If two matrices are congruent, they have the same eigenvalues. (c) Symmetric bilinear forms have symmetric matrix representations. (d) Any symmetric matrix is congruent to a diagonal matrix. (e) The sum of two symmetric bilinear forms is a symmetric bilinear form. (f) Two symmetric matrices with the same characteristic polynomial are matrix representations of the same bilinear form. (g) There exists a bilinear form 77 such that H(x,y) ^ 0 for all x and y-(h) If V is a vector space of dimension n, then dim(B(V)) = 2n. (i) Let 77 be a bilinear form on a finite-dimensional vector space V with dim(V) > 1. For any x G V, there exists y G V such that y^0,butH(x,y) = 0. (j) If 77 is any bilinear form on a finite-dimensional real inner product space V, then there exists an ordered basis 0 for V such that ijjp(H) is a diagonal matrix. 2. Prove properties 1, 2, 3, and 4 on page 423. 3. (a) Prove that the sum of two bilinear forms is a bilinear form. (b) Prove that the product of a scalar and a bilinear form is a bilinear form. (c) Prove Theorem 6.31. 4. Determine which of the mappings that follow are bilinear forms. Justify your answers. (a) Let V = C[0,1] be the space of continuous real-valued functions on the closed interval [0,1]. For / , g G V, define H(f,g)= f f(t)g(t)dt. Jo (b) Let V be a vector space over F , and let J G B(V) be nonzero. Define 77: V x V ^ F by 77(x, y) = [J(x, y)}2 for all x, y G V. 448 Chap. 6 Inner Product Spaces (c) Define 77: 77 x R -> R by II{h,t2) = *j + 2*2. (d) Consider the vectors of R2 as column vectors, and let 77: R2 * R be the function defined by H(x,y) = det(x,y), the determinant of the 2 x 2 matrix with columns x and y. (e) Let V be a real inner product space, and let 77: V x V 7? be the function defined by H(x,y) = (x.y) for x, y G V. (f) Let V be a complex inner product space, and let 7/: V x V * C be the function defined by H(x,y) = (x,y) for x.y G V. 5. Verify that each of the given mappings is a bilinear form. Then compute its matrix representation with respect to the given ordered basis 0. (a) H: R3 x R3 -> R, where = abi 2ab2 + a2^i 0363 6. 7. and (b) Let V = M2x2(77) and 0 = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (c) Define 77: V x V - R by 77(A B) = tr(A) tr(77). Let 0 = {cos*,sin*,cos2t,sin21}. Then 0 is an ordered basis for V = span(tf), a four-dimensional subspace of the space of all continuous functions on 7?. Let H: V x V defined by H(f,g) = f'(0)-g'(0). Let 77: R2 77 be the function // R be the function defined by = a i62 + a2b for G R' (a) Prove that 77 is a bilinear form. (b) Find the 2 x 2 matrix A such that 77(x. y) = x' Ay for all x, y G R2. For a 2 x 2 matrix A7 with columns x and y, the bilinear form 77(A7) = H(x,y) is called the pe rmanen t of M. Let V and W be vector spaces over the same field, and let T: V * W be a linear transformation. For any 77 G #(W), define f (77): V x V - F by f(H)(x,y) = H(T(x),T(y)) for all x,y G V. Prove the following results. Sec. 6.8 Bilinear and Quadratic Forms 449 (a) If 77 G B(W), then f (77) G (V). (b) T: #(W) > B(f) is a linear transformation. (c) If T is an isomorphism, then so is T. 8. Assume the notation of Theorem 6.32. (a) Prove that for any ordered basis 0, ipp is linear. (b) Let 0 be an ordered basis for an n-dimensional space V over F , and let 4>p: V * Fn be the standard representation of V with respect to 0. For A G M n X n(F) , define 77: V x V -> F by H(x,y) = [(f>0(x)tA[(j)fi{y)]. Prove that 77 G #(V). Can you establish this as a corollary to Exercise 7? (c) Prove the converse of (b): Let 77 be a bilinear form on V. If A = MH), then H(x,y) = [M^Y^Mv)]-9. (a) Prove Corollary 1 to Theorem 6.32. (b) For a finite-dimensional vector space V, describe a method for finding an ordered basis for & and 0 7 are the standard representations of V with respect to 0 and 7, respectively. (b) Apply Corollary 2 to Theorem 6.32 to (a) to obtain an alternative proof of Theorem 6.33. 14. Let V be a finite-dimensional vector space and 77 G #(V). Prove that, for any ordered bases 0 and 7 of V, rank(V^(77)) = rank(?^7(77)). 15. Prove the following results. (a) Any square diagonal matrix is symmetric. (b) Any matrix congruent to a diagonal matrix is symmetric. (c) the corollary to Theorem 6.35 16. Let V be a vector space over a field F not of characteristic two, and let 77 be a symmetric bilinear form on V. Prove that if K(x) = 77(x, x) is the quadratic form associated with 77, then, for all x, y G V, H(x,y) = -[K(x + y)-K(x)-K(y)}. 450 Chap. 6 Inner Product Spaces 17. For each of the given quadratic forms i ^ o n a real inner product space V, find a symmetric bilinear form 77 such that K(x) = 77(x, x) for all x G V. Then find an orthonormal basis 0 for V such that ipp(H) is a diagonal matrix. (a) K: R2 (b) K: R2 R defined by K ' t l * ' J ' ' *2. -2*^+4*1*2 + *| 77 defined by K r1 J = lt - 8*i*2 + t (u R defined by K *2 = 3*? + Zt + 3* - 2*i*3 W (c) K: R3 VW 18. Let S be the set of all (*i, i2, *3) G R3 for which 3*? + St2, + 3*1 - 2*1*3 + 2v^(*i + *3) + 1 = 0. Find an orthonormal basis 0 for R3 for which the equation relating the coordinates of points of S relative to 0 is simpler. Describe Sec. 6.9 Einstein's Special Theory of Relativity 451 22. For each of the following matrices A with entries from R, find a diagonal matrix D and an invertible matrix Q such that QlAQ = D. 24. 25. #m /or ^ : Use an elementary operation other than interchanging columns. 23. Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal matrix is congruent to the original one. Let T be a linear operator on a real inner product space V, and define H: V x V -> R by H(x, y) = (x, T(y)) for all x, y V. (a) Prove that H is a bilinear form. (b) Prove that H is symmetric if and only if T is self-adjoint. (c) What properties must T have for H to be an inner product on V? (d) Explain why H may fail to be a bilinear form if V is a complex inner product space. Prove the converse to Exercise 24(a): Let V be a finite-dimensional real inner product space, and let H be a bilinear form on V. Then there exists a unique linear operator T on V such that H(x, y) = (ar, T(y)) for all x, y e V , Hint: Choose an orthonormal basis (3 for V, let A = ipp(H), and let T be the linear operator on V such that [T]^ = A. Apply Exercise 8(c) of this section and Exercise 15 of Section 6.2 (p. 355). 26. Prove that the number of distinct equivalence classes of congruent nxn real symmetric matrices is (n + l)(n + 2) 6.9* EINSTEIN'S SPECIAL THEORY OF RELATIVITY As a consequence of physical experiments performed in the latter half of the nineteenth century (most notably the Michelson Morley experiment of 1887), physicists concluded that the results obtained in measuring the speed of light are independent of the velocity of the instrument used to measure the speed of light. For example, suppose that while on Earth, an experimenter measures the speed of light emitted from the sun and finds it to be 186,000 miles per second. Now suppose that the experimenter places the measuring equipment in a spaceship that leaves Earth traveling at 100,000 miles per second in a direction away from the sun. A repetition of the same experiment from the spaceship yields the same result: Light is traveling at 186,000 miles per second 452 Chap. 6 Inner Product Spaces relative to the spaceship, rather than 86,000 miles per second as one might expect! This revelation led to a new way of relating coordinate systems used to locate events in spacetime. The result was Albert Einstein's special theory of relativity. In this section, we develop via a linear algebra viewpoint the essence of Einstein's theory. S' Figure 6.8 The basic problem is to compare two different inertial (nonaccelerating) coordinate systems S and S' in three-space (R3) that are in motion relative to each other under the assumption that the speed of light is the same when measured in either system. We assume that S' moves at a constant velocity in relation to S as measured from S. (See Figure 6.8.) To simplify matters, let us suppose that the following conditions hold: 1. The corresponding axes of S and S' (x and x', y and y', z and z') are parallel, and the origin of S' moves in the positive direction of the x-axis of S at a constant velocity v > 0 relative to S. 2. Two clocks C and C are placed in spacethe first stationary relative to the coordinate system S and the second stationary relative to the coordinate system S'. These clocks are designed to give real numbers in units of seconds as readings. The clocks are calibrated so that at the instant the origins of S and S' coincide, both clocks give the reading zero. 3. The unit of length is the light second (the distance light travels in 1 second), and the unit of time is the second. Note that, with respect to these units, the speed of light is 1 light second per second. Given any event (something whose position and time of occurrence can be described), we may assign a set of space-time coordinates to it. For example, Sec. 6.9 Einstein's Special Theory of Relativity if p is an event that occurs at position 453 relative to S and at time t as read on clock C, we can assign to p the set of coordinates M y z W This ordered 4-tuple is called the spacetime coordinates of p relative to S and C. Likewise, p has a set of space-time coordinates (x' y z v7 relative to S' and C. For a fixed velocity v, let Jv: R4 T, /x y z W R4 be the mapping defined by fx' y z V) where (x y z and fx' y z are the spacetime coordinates of the same event with respect to 5 and C and with respect to S' and C', respectively. Einstein made certain assumptions about T that led to his special theory of relativity. We formulate an equivalent set of assumptions. Axioms of the Special Theory of Relativity (R 1) The speed of any light beam, when measured in either coordinate system using a clock stationary relative to that coordinate system, is 1. 454 (R 2) The mapping T: R1 (R 3) If Chap. 6 Inner Product Spaces R4 is an isomorphism. T, /x y (x' y z V) then y' = y and z' = z. (R.4) If T, (x l) -1 V) ,' - k u V and T r /x t ) = (x' y' z' V') then x' = x' and t' - t'. (R 5) The origin of S moves in the negative direction of the rr'-axis of Sf at the constant velocity v < 0 as measured from S'. Axioms (R 3) and (R 4) tell us that for p 6 R4, the second and third coor-dinates of T(p) are unchanged and the first and fourth coordinates of Tv(p) are independent of the second and third coordinates of p. As we will see, these five axioms completely characterize Tt,. The operator Tv is called the Lorentz t ransformat ion in direction :/;. We intend to compute T and use it to study the curious phenomenon of time contraction. Theorem 6.39. On R4, the following statements are true. (a) Tv(ei) = e-i for i = 2,3. (b) span({e2, e.3}) is T,,-invariant. (c) span({f'i,e4}) is Tv-invariant. (d) Both span({r;2lfi3}) and span({/?i,C4}) are T*-/nvariant. (e) T*v(('i) = et fori = 2,3. Proof, (a) By axiom (R 2), T and hence, by axiom (R 4), the first and fourth coordinates of () 0 0 w (o 0 0 w T, w V - 'Mf () 1 0 W (0 1 0 v ^ and Tv (0 0 1 vO> / 0 0 1 v^ y Sec. 6.9 Einstein's Special Theory of Relativity 455 are both zero for any a,b R. Thus, by axiom (R 3), T, The proofs of (b), (c), and (d) are left as exercises. (e) For any j / 2, (T*,(e2), ej) - (e2, T^e , ) ) = 0 by (a) and (c); for j = 2, (Tl{e2),ej) = 0 the light flash is observable from any point whose distance to the origin of S (as measured on S) is t 1 = t. These are precisely the points that lie on the sphere of radius t with center at the origin. The coordinates (relative to 456 Chap. 6 Inner Product Spaces S) of such points satisfy the equation x2 + y2 + z2 t2 0. Hence an event lies in P if and only if its space time coordinates [t > 0) relative to S and C satisfy the equation x2 + y2 + z2 ~ t2 = 0. By virtue of axiom (R 1), we can characterize P in terms of the space time coordinates relative; to S' and C similarly: An event lies in P if and only if, relative to S' and C, its space-time coordinates f.r' II w (t > 0) satisfy the equation {x')2 + (;/)2 + {z')2 - {t')2 = 0. Let A = / l 0 0 0 1 0 0 0 1 0 0 0 -l) Theorem 6.40. If (U(u>), w) - 0 for some w R4, then {VvLATv(w),w)=0. Proof. Let 'A w (I 'I GR4, and suppose that (LA(W).W) 0. CASE 1. t > 0. Since (L^w), w?) = x2 + y2 + z2 t2, the vector w gives the coordinates of an event in P relative to S and C. Because / .A and fx' y' Vl VJ Sec. 6.9 Einstein's Special Theory of Relativity 457 are the spacetime coordinates of the same event relative to S' and C, the discussion preceding Theorem 6.40 yields {x'Y + {y'Y + (z'Y - {t'Y = 0. Thus (T*vLATv(w),w) = (lATv(w),Tv(w)) = (x')2 + (y')2 + (z')2 - (t/)2 = 0, and the conclusion follows. CASE 2. t < 0. The proof follows by applying case 1 to w. 1 We now proceed to deduce information about Tv. Let wi = n 0 0 W and Wo. = ( W 0 0 v - v By Exercise 3, {wi,w2} is an orthogonal basis for span({ei,e4J), and span({ei,e4}) is T*L^Tv-invariant. The next result tells us even more. Theorem 6.41. There exist nonzero scalars a, and b such that (a) TlLATv(wi) = aw2. (b) T*vLATv(u)2) = bwi. Proof, (a) Because (-A(wi),w) 0, (TlLAlv{w),w1) = 0 by Theo-rem 6.40. Thus T*L^Tw(u'i) is orthogonal to w. Since span({ei,e4}) = span({w;i,tij2}) is T*LATv-invariant, T*-ATv{ivi) must lie in this set. But {wi,u>2} is an orthogonal basis for this subspace, and so T^Lj^Jv{w{) must be a multiple of w2- Thus T*L^Tu(w;1) = aw2 for some scalar a. Since Tv and A are invertible, so is T,*L^T. Thus a ^ 0, proving (a). The proof of (b) is similar to (a). I Corollary. Let Bv [Tv]0, where ft is the standard ordered basis for R4. Then (a) B*ABV = A. (b) T^L^T. = LA. We leave the proof of the corollary as an exercise. For hints, see Exercise 4. Now consider the situation 1 second after the origins of S and S' have coincided as measured by the clock C. Since the origin of S' is moving along the #-axis at a velocity v as measured in S, its space time coordinates relative to S and C are 458 Chap. 6 Inner Product Spaces Similarly, the spacetime coordinates for the origin of S' relative to S' and C must be / o 0 0 VI for some t' > 0. Thus we have T, fv 0 0 W / o 0 0 w for some f > 0. (18) By the corollary to Theorem 6.41, T * L A T (v 0 0 VI i (v 0 0 VI = (LA fv 0 0 VI ) fv 0 0 VI = v2-l. (19) But also T ; L A T W /v 0 0 VI 5 (v 0 0 VI = (LATV = u / t l 0 0 VI ,TV fv 0 0 VI / o 0 0 VI i (0 0 0 VI = -(t f2 (20) Combining (19) and (20), we conclude that v 1 = (f), or t ' = VT - u * . (21) Thus, from (18) and (21), we obtain T, M o 0 w / o 0 0 Vi=^I (22) Next recall that the origin of S moves in the negative direction of the x'-axis of S' at the constant velocity v < 0 as measured from S'. [This fact Sec. 6.9 Einstein's Special Theory of Relativity 459 is axiom (R 5).] Consequently, 1 second after the origins of S and S' have coincided as measured on clock C, there exists a time t' > 0 as measured on clock C such that T, / 0 0 0 W f-vt' 0 0 V ' ' / (23) From (23), it follows in a manner similar to the derivation of (22) that 1 t' = v T ^ 2 ' (24) hence, from (23) and (24), T, / 0 0 0 VI / -v Vl-v2 0 0 1 (25) WT^v2! The following result is now easily proved using (22), (25), and Theorem 6.39. Theorem 6.42. Let 0 be the standard ordered basis for R4. Then { 1 Pwjfl 'v o 0 v VT^ 0 0 1 0 0 1 0 0 VT^ 0 1 Time Contraction A most curious and paradoxical conclusion follows if we accept Einstein's theory. Suppose that an astronaut leaves our solar system in a space vehicle traveling at a fixed velocity v as measured relative to our solar system. It follows from Einstein's theory that, at the end of time t as measured on Earth, the time that passes on the space vehicle is only t/ v2. To establish this result, consider the coordinate systems S and S' and clocks C and C that we have been studying. Suppose that the origin of S' coincides with the space vehicle and the origin of S coincides with a point in the solar system 460 Chap. 6 Inner Product Spaces (stationary relative to the sun) so that the origins of 5' and S' coincide and clocks C and C read zero at the moment the astronaut embarks on the trip. As viewed from 5, the space-time coordinates of the vehicle at any time t > 0 as measured by C are /vt 0 0 VI whereas, as viewed from S', the space-time coordinates of the vehicle at any time f > 0 as measured by C are / 0 0 0 VI But if two sets of space -time coordinates M 0 0 VI and 0 0 VI are to describe the same event, it must follow that T, fvt 0 0 VI (o 0 0 VI Thus L' vft &v ( 1 Vl-v2 0 0 v 1 0 0 0 1 0 VvT V VT^ o I /vt / 0 0 0 V ^ 0 0 VI From the preceding equation, we obtain -vH + v 7 ! 3 ^ VT^ = f, or t' = ty/l-v2. (26) Sec. 6.9 Einstein's Special Theory of Relativity 461 This is the desired result. A dramatic consequence of time contraction is that distances are con-tracted along the line of motion (see Exercise 9). Let us make one additional point. Suppose that we consider units of distance and time more commonly used than the light second and second, such as the mile and hour, or the kilometer and second. Let c denote the speed of light relative to our chosen units of distance. It is easily seen that if an object travels at a velocity v relative to a set of units, then it is traveling at a velocity v/c in units of light seconds per second. Thus, for an arbitrary set of units of distance and time, (26) becomes f = , / i - ? . EXERCISES 1. Prove (b), (c), and (d) of Theorem 6.39. 2. Complete the proof of Theorem 6.40 for the case t < 0. 3. For W = show that (a) {w,W2} is an orthogonal basis for span({ei,C4}); (b) span({ei,e4}) is T*LATw-invariant. 4. Prove the corollary to Theorem 6.41. Hints: fi 0 0 w and u)2 ( l 0 0 v - v (a) Prove that B*ABV = (p 0 0 Vi 0 1 0 0 0 0 1 0 0 0 -p) where a + b a b p = - and q = -. 462 Chap. 6 Inner Product Spaces w = (b) Show that q 0 by using the fact that B*ABV is self-adjoint. (c) Apply Theorem 6.40 to M i o VI to show that p = 1. 5. Derive (24). and prove that T f{)) 0 0 V'J ' s/l-v* 0 0 1 (25) Xs/T^I Hint: Use a technique similar to the derivation of (22). 6. Consider three coordinate systems S. S', and S' with the corresponding axes (j;,:/;',:/;'; y,y',y': and z,z'.z') parallel and such that the x-, x'-. and :r'-axes coincide. Suppose that S' is moving past S at a velocity t'i > 0 (as measured on S). S' is moving past S' at a velocity u2 > 0 (as measured on 5'), and S' is moving past S at a velocity V3 > 0 (as measured on .9), and that there are three clocks C. C, and C' such that C is stationary relative to S, C is stationary relative to S', and C' is stationary relative to S'. Suppose; that when measured on any of the three clocks, all the origins of S, S'. and S' coincide at time 0. Assuming that Tr., = T2T,., (i.e.. BV.A = BV2BVl), prove that V3 = Note that substituting v2 = 1 in this equation yields V3 = 1. This tells is that the speed of light as measured in 5 or S' is the same. Why would we be surprised if this were not the case? 7. Compute (By)'1. Show ( B r ) _ l = B{ r). Conclude that if S' moves at a negative velocity v relative to S. then [T,.] ,^ = B,.. where B,. is of the form given in Theorem 6.42. 8. Suppose that an astronaut left Earth in the year 2000 and traveled to a star 99 light years away from Earth at 99% of the speed of light and that upon reaching the star immediately turned around and returned to Earth at the same speed. Assuming Einstein's special theory of Sec. 6.9 Einstein's Special Theory of Relativity 463 relativity, show that if the astronaut was 20 years old at the time of departure, then he or she would return to Earth at age 48.2 in the year 2200. Explain the use of Exercise 7 in solving this problem. 9. Recall the moving space vehicle considered in the study of time contrac-tion. Suppose that the vehicle is moving toward a fixed star located on the :r-axis of 5 at a distance b units from the origin of S. If the space vehicle moves toward the star at velocity v, Earthlings (who remain 'al-most' stationary relative to S) compute the time it takes for the vehicle to reach the star as t = b/v. Due to the phenomenon of time contraction, the astronaut perceives a time span of f = ty/1 v2 = (b/v)y/l v2. A paradox appears in that the astronaut perceives a time span incon-sistent with a distance of b and a velocity of v. The paradox is resolved by observing that the distance from the solar system to the star as measured by the astronaut is less than b. Assuming that the coordinate systems S and S' and clocks C and C are as in the discussion of time contraction, prove the following results. (a) At time t (as measured on C), the space time coordinates of star relative to S and C are fb 0 0 VI (b) At time t (as measured on C), the spacetime coordinates of the star relative to S' and C are / b - vt 0 0 t-bv V/T^I (c) For , b tv . t bv x = , and t = VT=r v T ^ ' we have x' = by/1 v2 fv. This result may be interpreted to mean that at time t! as measured by the astronaut, the distance from the astronaut to the star as measured by the astronaut (see Figure 6.9) is bJ - v2 - fv. 464 Chap. 6 Inner Product Spaces S' Figure 6.9 (x',0,0) coordinates relative to S' *(star) (6,0,0) coordinates relative to S (d) Conclude from the preceding equation that (1) the speed of the space vehicle relative to the star, as measured by the astronaut, is v, (2) the distance from Earth to the star, as measured by the astro-naut, is b/ v2. Thus distances along the line of motion of the space vehicle appear to be contracted by a factor of s/l v2. 6.10* CONDITIONING AND THE RAYLEIGH QUOTIENT In Section 3.4, we studied specific techniques that allow us to solve systems of linear equations in the form Ax b. where A is an m x n matrix and 6 is an m x 1 vector. Such systems often arise in applications to the real world. The coefficients in the system are frequently obtained from experimental data, and. in many cases, both m and n are so large that a computer must be used in the calculation of the solution. Thus two types of errors must be considered. First, experimental errors arise; in the collection of data since no instruments can provide completely accurate measurements. Second, computers introduce roundoff errors. One might intuitively feel that small relative changes in the coefficients of the system cause; small relative errors in the solution. A system that has this property is called well-conditioned: otherwise, the system is called ill-conditioned. We now consider several examples of these types of errors, concentrating primarily on changes in 6 rather than on changes in the entries of A. In addition, we assume that A is a square;, complex (or real), invcrtible matrix since this is the case most freemently encountereel in applications. i Sec. 6.10 Conditioning and the Rayleigh Quotient 465 Example 1 Consider the system The solution to this system is x + x2 = 5 X X2 1. 2 / -Now suppose that we change the system somewhat and consider the new system x + x2 = 5 xi - x2 = 1.0001. This modified system has the solution 3.00005 1.99995/ ' We see that a change of 10 4 in one coefficient has caused a change of less than 10 - 4 in each coordinate of the new solution. More generally, the system $i + X2 = 5 X X2 = 1 4- 5 has the solution 3 + 6/2 2-8/2 Hence small changes in b introduce small changes in the solution. Of course, we are really interested in relative changes since a change in the solution of, say, 10, is considered large if the original solution is of the order 10 - 2 , but small if the original solution is of the order 106. We use the notation 5b to denote the vector b' b, where b is the vector in the original system and b' is the vector in the modified system. Thus we have 8b = We now define the relative change in b to be the scalar ||o'6||/||&||, where || || denotes the standard norm on Cn (or Rn); that is, ||6|| = y/(b,b). Most 466 Chap. 6 Inner Product Spaces of what follows, however, is true for any norm. Similar definitions holel for the relative change in x. In this example, I N '26 and 6x (3 + (h/2) 2 ~ (h/2)J 0 - G ) '26 Thus the relative change in x equals, coincidentally, the relative change in b; se> the system is well-conditioned. Example 2 Consider the system X + X2 = 3 X + 1.00001a-2 = 3.00001, which has as its solution. The solution to the relateel system x + x2 = 3 X] + 1.00001x2 = 3.00001 + e5 is Hence, while1 llfall ll-rll 2 - (105)//' i + (io5)/.y = 105>/275|/i| > 104|//.|, l l ^ l Zy/2 Thus the relative change in x is at least 104 times the relative change in 6! This system is very ill-conditioned. Observe; that the lines defineel by the two equations are nearly coincident. Se> a small change in either line could greatly alter the point of intersection, that is, the solution to the system. Sec. 6.10 Conditioning and the Rayleigh Quotient 467 To apply the full strength of the theory of self-adjoint matrices to the study of conditioning, we need the notion of the norm of a matrix. (See Exercise 24 of Section 6.1 for further results about norms.) Definition. Let A be a complex (or real) n x n matrix. Define the (Euclidean) norm of A by A = max Ax xjtO x where x G Cn or x G Rn. Intuitively, A represents the maximum magnification of a vector by the matrix A. The question of whether or not this maximum exists, as well as the problem of how to compute it, can be answered by the use of the so-called Rayleigh quotient. Definition. Let B be an n x n self-adjoint matrix. The Rayleigh quotient for x ^ 0 is defined to be the scalar R(x) = (Bx,x) / | |x| |2 . The following result characterizes the extreme values of the Rayleigh quo-tient of a self-adjoint matrix. Theorem 6.43. For a self-adjoint matrix B G MnXn(F), we have that max R(x) is the largest eigenvalue ofB and min R(x) is the smallest eigenvalue x^O xj^O ofB. Proof. By Theorems 6.19 (p. 384) and 6.20 (p. 384), we may choose an orthonormal basis {vi, v 2 , . . . , vn} of eigenvectors of B such that Bvi = Ajfj (1 < i < n), where Ai > A2 > > ATO. (Recall that by the lemma to Theorem 6.17, p. 373, the eigenvalues of B are real.) Now, for x G Fn, there exist scalars oi, a 2 , . . . , an such that Hence R(x) = X = ^TaiVi. i=l BX, X) E'=1 aiXiVi' EJ=1 aJV> I j - | >' ~~ II T* I ^ It is easy to see that R(vi) = Aj, so we have demonstrated the first half of the theorem. The second half is proved similarly. H 468 Chap. 6 Inner Product Spaces Corollary 1. For any square matrix A, A is finite and, in fact, equals y/X, where A is the largest eigenvalue of A* A. Proof. Let B be the self-adjoint matrix A*A, and let A be the largest eigenvalue of B. Since, for x / 0, 0 < |Ar| |2 (Ax, Ax) (A*Ax,x) it follows from Theorem 6.43 that p | | 2 = A. x 2 _ < # - * > . Observe that the proof of Corollary 1 shows that all the eigenvalues of A* A are nonnegative. For our next result, we need the following lemma. Lemma. For any square matrix A, A is an eigenvalue of A* A if and only if A is an eigenvalue of AA*. Proof. Let A be an eigenvalue of A* A. If A = 0, then A* A is not invertible. Hence A and A* are not invertible, so that A is also an eigenvalue of AA*. The proof of the converse is similar. Suppose now that A^O. Then there exists x ^ 0 such that A* Ax = x. Apply A to both sides to obtain (AA*)(Ax) = (Ax). Since Ax ^ 0 (lest Xx = 0), we have that A is an eigenvalue of AA*. The proof of the converse is left as an exercise. I Corollary 2. Let A be an invertible matrix. Then ||J4_1|| = l / /A, where A is the smallest eigenvalue of A* A. Proof. Recall that A is an eigenvalue of an invertible matrix if and only if A - 1 is an eigenvalue of its inverse. Ne>w let Ai > A2 > > ATl be the eigenvalues of A* A, which by the lemma are the eigenvalues of AA*. Then A~~1 2 equals the largest eigenvalue of (A'1)*A'1 = (AA*)~ which equals 1/An. I For many applications, it is only the largest and smallest eigenvalues that are of interest. For example, in the case of vibration problems, the smallest eigenvalue represents the lowest frequency at which vibrations can occur. We see the role of both of these eigenvalues in our study of conditioning. Example 3 Let Sec. 6.10 Conditioning and the Rayleigh Quotient Then 469 The eigenvalues of B are 3, 3, and 0. Therefore, ||^4|| = /3- For any x=b^0, we may compute R(x) for the matrix B as (Bx, x) 2(a2 + b2 + c2 -ab + ac + be) 3 > R(x) = a2 + b2-r c2 Now that we know ||^4|| exists for every square matrix A, we can make use of the inequality Ax < A x, which holds for every x. Assume in what follows that A is invertible, b ^ 0, and Ax = b. For a given 8b, let 5x be the vector that satisfies A(x + 8x) = b + 8b. Then A(8x) 8b, and so 8x = A-1 (8b). Hence - Ax < A x and 8x = H ^ W I I < P ' II W. Thus 8x < < WA-'w-im-wAw i m i I I& I I /P I I Similarly (see Exercise 9), b = , , . - . ( ) . (b^ Sx A-A-i bJ < The number A ||>1-1|| is called the condition number of A and is denoted cond(yl). It should be noted that the definition of cond(^4) depends on how the norm of A is defined. There are many reasonable ways of defining the norm of a matrix. In fact, the only property needed to establish the inequalities above is that Ax < A x for all x. We summarize these results in the following theorem. Theorem 6.44. For the system Ax b, where A is invertible and b ^ 0, the following statements are true. 1 5b (a) For any norm || ||, we have cond(A) ll470 Chap. 6 Inner Product Spaces (b) If || || is the Euclidean norm, then cond(A) = /Ai/An , where Ai and An are the largest and smallest eigenvalues, respectively, of A* A. Proof. Statement (a) follows from the previous inequalities, and (b) follows from Corollaries 1 and 2 to Theorem 6.43. 1 It is clear from Theorem 6.44 that cond(^4) > 1. It is left as an exercise to prove that cond(A) = 1 if and only if A is a scalar multiple of a unitary or orthogonal matrix. Moreover, it can be shown with some work that equality can be obtained in (a) by an appropriate choice of b and 8b. We can see immediately from (a) that if cond(^4) is close to 1, then a small relative error in b forces a small relative error in x. If cond(A) is large, however, then the relative error in x may be small even though the relative error in b is large, or the relative error in x may be large even though the relative error in b is small! In short, cond(^4) merely indicates the potential for large relative errors. We have so far considered only errors in the vector b. If there is an error 8A in the coefficient matrix of the system Ax b, the situation is more complicated. For example, A + 8A may fail to be invertible. But under the appropriate assumptions, it can be shown that a bound for the relative error in x can be given in terms of cond(A). For example, Charles Cullen (Charles G. Cullen, An Introduction to Numerical Linear Algebra, PWS Publishing Co., Boston 1994, p. 60) shows that if A + 8A is invertible, then 5, 8x < cond(.4) 8A JM It should be mentioned that, in practice, one never computes cond(v4) from its definition, for it would be an unnecessary waste of time to compute A-1 merely to determine its norm. In fact, if a computer is used to find A~l, the computed inverse of A in all likelihood only approximates A~l, and the error in the computed inverse is affected by the size of cond(A). So we are caught in a vicious circle! There are, however, some situations in which a usable approximation of cond(^4) can be found. Thus, in most cases, the estimate of the relative error in x is based on an estimate of cond(^4). EXERCISES 1. Label the following statements as true or false. (a) If Ax = b is well-conditioned, then cond(^4) is small. (b) If cond(A) is large, then Ax = b is ill-conditioned. (c) If cond(^4) is small, then Ax b is well-conditioned. (d) The norm of A equals the Rayleigh quotient. (e) The norm of A always equals the largest eigenvalue of A. Sec. 6.10 Conditioning and the Rayleigh Quotient 471 2. Compute the norms of the following matrices, (a) 4 0 1 s) (b) ( - i i ) (e) f1 % 0 Tz l 75 j 3. Prove that if B is symmetric, then ||Z?|| is the largest eigenvalue of B. 4. Let A and A-1 be as follows: / 6 13 -vr A = 13 29 - 3 8 V17 - 3 8 50, and A l = The eigenvalues of A are approximately 84.74, 0.2007, and 0.0588. (a) Approximate ||i4||, | |A - 1 | | , and cond(^l). (Note Exercise 3.) (b) Suppose that we have vectors x and x such that Ax = b and ||6 Ax < 0.001. Use (a) to determine upper bounds for x - A~lb (the absolute error) and x - .4-16||/||.A_1&|| (the rel-ative error). 5. Suppose that x is the actual solution of Ax = b and that a computer arrives at an approximate solution x. If cond(J4) = 100, ||6|| = 1, and ||6 Ax = 0.1, obtain upper and lower bounds for x x||/| |x||. 6. Let B = Compute | B ||, and cond(Z?). 7. Let B b e a symmetric matrix. Prove that min R(x) equals the smallest x^O eigenvalue of B. 8. Prove that if A is an eigenvalue of AA*, then A is an eigenvalue of A* A. This completes the proof of the lemma to Corollary 2 to Theorem 6.43. 9. Prove that if A is an invertible matrix and Ax b, then i / -8b. . 8x A-A - i i n m < v II&II J -472 Chap. 6 Inner Product Spaces 10. Prove the left inequality of (a) in Theorem 6.44. 11. Prove that cond(.4) = 1 if and only if A is a scalar multiple of a unitary or orthogonal matrix. 12. (a) Let A and B be square matrices that are unitarily equivalent. Prove that A = B. (b) Let T be a linear operator on a finite-dimensional inner product space V. Define |T|| = max T(x) Prove that ||T|| = ||[T]^||,.where (3 is any orthonormal basis for V. (c) Let V be an infinite-dimensional inner product space with an or-thonormal basis {vi,v2,...}. Let T be the linear operator on V such that T(vk) = kvk- Prove that ||T|| (defined in (b)) does not exist. The next exercise assumes the definitions of singular value and pseudoinverse and the results of Section 6.7. 13. Let A be an n x n matrix e)f rank r with the nonzero singular values o~i > cr2 > > rrr. Prove each of the following results. (a) | | A | | = a i . (b) ||At|| = i . (c) If A is invertible (and hence r = n), then cond(vl) = . o~ 6.11* THE GEOMETRY OF ORTHOGONAL OPERATORS By Theorem 6.22 (p. 386), any rigid motion on a finite-dimensional real inner product space is the composite of an orthogonal operator and a translation. Thus, to understand the geometry of rigid motions thoroughly, we must ana-lyze the structure of orthogonal ejperators. Such is the aim of this section. We show that any orthogonal operator on a finite-dimensional real inner product space is the composite of rotations and reflections. This material assumes familiarity with the results about direct sums de-veloped at the end of Section 5.2, and familiarity with the definition and elementary properties of the determinant of a linear operator defined in Ex-ercise 7 of Section 5.1. Definitions. Let T be a linear operator on a finite-dimensional real inner product space V. The operator T is called a rotation if T is the identity on Sec. 6.11 The Geometry of Orthogonal Operators 473 V or if there exists a two-dimensional subspace W ol'V, an orthonormal basis (3 = {xi, x2} ^or W, and a real number 0 such that T(xi) = (cos0)xi + (sin0)x2, T(x2) = (sint9)xi + (cos0)x2, and T(y) y for all y G N. In this context, T is called a rotation of W about W'1. The subspace W-1 is called the axis of rotation. Rotations are defined in Section 2.1 for the special case that V = R2. Definitions. Let T be a linear operator on a finite-dimensional real inner product space V. The operator T is called a reflection if there exists a one-dimensional subspace W of V such that T(x) = x for all x G W and T(y) = y for all y G W-1. In this context, T is called a reflection ofV about W 1 . It should be noted that rotations and reflections (or composites of these) are orthogonal operators (see Exercise 2). The principal aim of this section is to establish that the converse is also true, that is, any orthogonal operates on a finite-dimensional real inner product space is the composite of rotations and reflections. Example 1 A Characterization of Orthogonal Operators on a One-Dimensional Real In-ner Product Space Let T be an orthogonal operator on a one-dimensional inner product space V. Choose any nonzero vector x in V. Then V = span({x}), and so T(x) = Ax for some A G R. Since T is orthogonal and A is an eigenvalue of T, A = 1. If A = 1, then T is the identity on V, and hence T is a rotation. If A = 1, then T(x) x for all x G V; so T is a reflection of V about V1- = {0}. Thus T is either a rotation or a reflection. Note that in the first case, det(T) = 1, and in the second case, det(T) = 1. Example 2 Some Typical Reflections (a) Define T: R2 -> R2 by T(a,b) = (-0,6) , and let W = span({ei}). Then T(x) = -x for all x G W, and T(y) = y for all 1 / 6 W 1 . Thus T is a reflection of R2 about W1 = span({e2}), the y-axis. (b) Let T: R3 - R3 be defined by T(a,b,c) = (a,b,-c), and let W = span({e3}). Then T(x) = - x for all x G W, and T(y) - y for all y W'1 = span({ei,e2}), the xy-plane. Hence T is a reflection of R3 about W-1-. Example 1 characterizes all orthogonal operators on a one-dimensional real inner product space. The following theorem characterizes all orthogonal 474 Chap. 6 Inner Product Spaces operators on a two-dimensional real inner product space V. The proof fol-lows from Theorem 6.23 (p. 387) since all two-dimensional real inner product spaces are structurally identical. For a rigorous justification, apply Theo-rem 2.21 (p. 104), where j3 is an orthonormal basis for V. By Exercise 15 of Section 6.2, the resulting isomorphism (j>p: V R2 preserves inner products. (See Exercise 8.) Theorem 6.45. Let T he an orthogonal operator on a two-dimensional real inner product space V. Then T is either a rotation or a reflection. Fur-thermore, T is a rotation if and only if det(T) = 1, and T is a reflection if and only if det(T) = 1. A complete description of the reflections of R2 is given in Section 6.5. Corollary. Let V be a two-dimensional real inner product space. The composite of a reflection and a rotation on V is a reflection on V. Proof. If Ti is a reflection on V and T2 is a rotation on V, then by Theorem 6.45, det(Ti) = 1 and det(T2) = - 1 . Let T = T2Tj be the composite. Since T2 and Ti are orthogonal, so is T. Moreover, det(T) = det(T2) det(Ti) = 1. Thus, by Theorem 6.45, T is a reflection. The proof for TiT2 is similar. (S| We now study orthogonal operators on spaces of higher dimension. Lemma. If T is a linear operator on a nonzero finite-dimensional real vector space V, then there exists a T-invariant subspace W of V such that 1 < dim(W) < 2. Proof. Fix an ordered basis (3 = {y, y2, , 2/n} for V, and let A = [T]p. Let p: V > Rn be the linear transformation defined by p(yi) e% for i 1 ,2 , . . . ,n. Then p is an isomorphism, and, as we have seen in Sec-tion 2.4, the diagram in Figure 6.10 commutes, that is, LA(f>p = p. As a consequence, it suffices to show that there exists an L^-invariant subspace Z of Rn such that 1 < dim(Z) < 2. If we then define W = 4>^l(Z), it follows that W satisfies the conclusions of the lemma (see Exercise 13). V V Rn L^ , Rn Figure 6.10 Sec. 6.11 The Geometry of Orthogonal Operators 475 The matrix A can be considered as an n x n matrix over C and, as such, can be used to define a linear operator U on Cn by U(i>) = Av. Since U is a linear operator on a finite-dimensional vector space over C, it has an eigenvalue A G C. Let x G Cn be an eigenvector corresponding to A. We may write A = Ai + iA2, where Ai and A2 are real, and lax +ih a2 + ib2 x = an + ibn) where the a^'s and tVs are real. Thus, setting Xi = far Vn) and x2 = b2 Vn) we have x = X + ix2, where xi and x2 have real entries. Note that at least one of Xi or x2 is nonzero since x ^ 0. Hence U(x) = Ax = (Ai -I- ?'A2)(xi + 1x2) = (X A2x2) + i(X2 4- A2xi). Similarly, U(x) = ^4(xi + 1x2) = Ax + iAx2-Comparing the real and imaginary parts of these two expressions for U(x), we conclude that Ax 1 A1X1 A2x2 and Ax2 Ajx2 + A2xi. Finally, let Z = span({xi,x2}), the span being taken as a subspace of Rn. Since xi ^ 0 or x2 7^ 0, Z is a nonzero subspace. Thus 1 < dim(Z) < 2, and the preceding pair of equations shows that Z is L^-invariant. II Theorem 6.46. Let T be an orthogonal operator on a nonzero finite-dimensional real inner product space V. Then there exists a collection of pairwise orthogonal T-invariant subspaces {Wi, W 2 , . . . , WTO} ofV such that (a) 1 < dim(Wi) 1. 476 Chap. 6 Inner Product Spaces Suppose elim(V) = n. By the lemma, there exists a T-invariant subspace Wi of V such that 1 < dim(W) < 2. If Wi = V, the result is established. Otherwise, Nj- / {0}. By Exercise 14, Wj1 is T-invariant and the restriction of T to Wf is orthogonal. Since dim(Wf) < n, we may apply the induc-tion hypothesis to T w i and conclude that there exists a collection of pair-wise orthogonal T-invariant subspaces {Wi, W 2 , . . . , Wm} of W -^ such that 1 < dim(Wr a reflection for each i = 2 , 3 , . . . , m . Thus, in some sense, T is composed of rotations and reflections. Unfortunately, very little can be said about the uniqueness e)f the decomposition of V in Theorem 6.46. For example, the NA/j's, the number rn of Wj's, and the number of W,'s for which Tw, is a reflection are not unique. Although the number of W,'s for which Tw, is a reflection is not unique, whether this number is even or odd is an intrinsic property of T. Moreover, we can always decompose V so that Tw, is a reflection for at most one W,. These fae;ts are established in the following result. Theo rem 6.47. Let T, V, Wi, . . . . W, be as in Theorem 0.46. (a) The number ofWi s for which Tw, is a reflection is even or odd according to whether det(T) = 1 or det(T) = - 1 . (b) It is always possible to decompose V as in Theorem 6.46 so that the number of Wj s for which Tw, is a reflection is zero or one according to whether elet(T) = 1 or dct(T) = 1. Furthermore, i/Tw, is a reflection, then dim(Wi) = 1. Proof, (a) Let r denote the number of W,'s in the decomposition for which Twi is a reflection. Then, by Exercise 15, det(T) = det(TW l)- det(Tw , ) dct(T w, = ( - 1 proving (a). (b) Let E = {x G V: T(x) = x}; then E is a T-invariant subspace of V. If W = E , then W is T-invariant. So by applying Theorem 6.46 to Tw- we obtain a collection of pairwise orthogonal T-invariant subspaces {Wi, W 2 , . . . , Wfc} of W such that W = Wi 0 W2 0 e Wfc and for 1 < i < k, the dimension of each Wi is either 1 or 2. Observe that, for each i 1 ,2 , . . . ,k, Tw; is a rotation. For otherwise, if Tw, is a reflection, there; exists a nonzero x G W^ for which T(x) = x. But then, x G W, fi E C E 1 n E = {0}, a contradiction. If E = {()}, the result follows. Otherwise, Sec. 6.11 The Geometry of Orthogonal Operators 477 choose an orthonormal basis f3 for E containing p vectors (p > 0). It is possible to decompose ft into a pairwise disjoint union (3 = (3 U (32 U U (3r such that each /% contains exactly two vectors for i < r, and (3r contains two vectors if p is even and one vector if p is odd. For each i = 1, 2 , . . . , r, let Wk+i = span(/?i). Then, clearly, {Wi, W2, , W f c , . . . , W fc+r} is pairwise orthogonal, and V = Wi 0 W2 0 Wfc 0 Wfc+r Moreover, if any $ contains two vectors, then - 1 0 (27) det(TWfc+i) = det([T w. fii = det 0 - 1 = 1. So Twfc+i is a rotation, and hence Tw., is a rotation for j < k + r. If (3r consists of one vector, then dim(Wfc+r) = 1 and = det([T w. J A .) = d e t ( - l ) = - 1 . Thus Twfc+J. is a reflection by Theorem 6.46, and we conclude that the de-composition in (27) satisfies the condition of (b). | As a consequence of the preceding theorem, an orthogonal operator can be factored as a product of rotations and reflections. Corollary. Let T be an orthogonal operator on a finite-dimensional real inner product space V. Then there exists a collection {Ti ,T 2 , . . . , T m } of orthogonal operators on V such that the following statements are true. (a) For each i, T, is either a reflection or a rotation. (b) For at most one i, T% is a reflection. (c) TiTj = TjT; for all i and j. (d) T = T i T 2 - - - T m . (e) det(T) = 1 if Ti is a rotation for each i 1 otherwise. Proof. As in the proof of Theorem 6.47(b), we can write V = W 1 0 W 2 0 . . - 0 W m , where Tw, is a rotation for i < m. For each i = 1 ,2 , . . . , m, define T;: V + V by Tj(xi + x 2 + = Xi+X2- r- Xi-i + T(Xi) + X i+i -I 1- x m , where Xj G Wj for all j. It is easily shown that each T$ is an orthogonal operator on V. In fact, T; is a rotation or a reflection according to whether Tw; is a rotation or a reflection. This establishes (a) and (b). The proofs of (c), (d), and (e) are left as exercises. (See Exercise 16.) 1 478 Chap. 6 Inner Product Spaces Example 3 Orthogonal Operators on a Three-Dimensional Real Inner Product Space Let T be an orthogonal operator em a three-dimensional real inner product space V. We show that T can be decomposed into the composite of a rotation and at most one reflection. Let V = W , W 2 0 - - - 0 W m be a decomposition as in Theorem 6.47(b). Clearly, m = 2 or m = 3. If m = 2, then V = Wi W2. Without loss of generality, suppose that dim(Wi) = 1 and dim(W2) = 2. Thus Tw, is a reflection or the identity on Wi, and Tw2 Is a rotation. Defining Tj and T2 as in the proof of the corollary to Theorem 6.47, we have that T = TiT2 is the composite of a rotation and at most one reflection. (Note that if Tw, is not a reflection, then Ti is the identity on V and T = T2.) If m = 3, then V = Wx W2 W3 and ehm(W,) = 1 for all i. For each i, let Tj be as in the; proof of the corollary to Theorem 6.47. If Tw, is not a reflection, then Tj is the identity on Wj. Otherwise, Tj is a reflection. Since Twi is a reflection for at most one i, we conclude that T is either a single reflection or the identity (a rotation). EXERCISES 1. Label the following statements as true or false. Assume that the under-lying vector spaces are finite-dimensional real inner product spaces. (a) Any orthogonal operator is either a rotation or a re-flection. (b) The composite of any two rotations on a two-dimensional space is a rotation. (c) The composite e>f any two rotations on a three-dimensional space is a rotation. (d) The composite of any two rotations on a four-dimensional space is a rotation. (e) The ielentity operator is a rotation. (f) The composite of two reflections is a reflection. (g) Any orthogonal operator is a composite of rotations. (h) For any orthogonal operator T, if det(T) = 1, then T is a reflec-tion, (i) Reflections always have eigenvalues, (j) Rotations always have eigenvalues. 2. Prove that rotations, reflections, anel composites of rotations and re%-flections are orthogonal operators. ^ - T * i . Sec. 6.11 The Geometry of Orthogonal Operators 3. Let 479 A = ( I 2 v/3 /3 2 1 '2 I and B 0 - 1 (a) Prove that VA is a reflection. (b) Find the axis in R2 about which L^ reflects, that is, the subspace of R2 on which L^ acts as the identity. (c) Prove that -AB and -BA are rotations. 4. For any real number , let A = cos* U sm< sin* cos* (a) Prove that LA is a reflection. (b) Find the axis in R2 about which L^ reflects. 5. For any real number 480 Chap. 6 Inner Product Spaces 10. Prove that if V is a two- or three-dimensional real inner product space', then the cemiposite e>f two reflcctiems on V is a rotation of V. 11. Give an example of an orthogonal operator that is neither a rcfiection nor a rotation. 12. Lt;t V be a finite-dimensional real inner product spae-e. Define T: V * V by T(x) = x. Prove that T is a product e>f rotatiems if and only if dim(V) is even. 13. Complete the proof e>f the lemma to Theorem 6.46 by shewing that W = 0~i1(Z) satisfies the required conelitieais. 14. Let T be an orthogonal [unitary] operator on a finite-dimensional real [complex] inner product space V. If W is a T-invariant subspace erf V, prove the following rosults. (a) Tw is an orthogonal [unitary] operator on W. (b) WL is a T-invariant subspace of V. Hint: Use the fact that. Tw is one-to-one and onto te) concluele that, for any y G W. T*(y) = T '(,(/) G W. (c) T w is an orthogonal [unitary] operator on W. 15. Let T be1 a linear e)perate>r on a finite-eliniensional vector space V, where V is a diroct sum of T-invariant subspae-e;s, say, V = W] W 2 0- -cOW/,.. Prove that det(T) = elet(TW|) ele>t(Tw,,) ele>t(TwJ. 16. Complete the proof of the1 corollary to The'orcm 6.47. 17. Let T be' a linear e)perate>r on an n-dimensional real inner product space V. Suppe)se that T is not the identity. Prove the following results. (a) If a, is odd, then T can be e'xpressexl as the1 e-emiposite of at most one reflection anel at most (n 1) rotations. (b) If /? is even, then T can be1 e'xpresse>el as the' colllpe)site, of at most | n rotations e>r as the ce)iupe)sitc of one inflection and at most ^(n 2) rotations. 18. Let V be a real inner product spae*? e)f elimension 2. For any x, y G V sue:h that x / y and ||x|| = y = 1, show that there exists a unique rotation T em V such that T(x) y. INDEX OF DEFINITIONS FOR CHAPTER 6 Adjoint of a linear operator 358 Adjoint of a matrix 331 Axis of rotation 473 Bilinear form 422 Complex inner product space 332 Condition number 469 Chap. 6 Index of Definitions 481 Congruent matrices 426 Conjugate transpose (adjoint) of a matrix 331 Critical point 439 Diagonalizable bilinear form 428 Fourier coefficients of a vector rela-tive to an orthonormal set 348 Frobenius inner product 332 Gram-Schmidt orthogonalization process 344 Hessian matrix 440 Index of a bilinear form 444 Index of a matrix 445 Inner product 329 Inner product space 332 Invariants of a bilinear form 444 Invariants of a matrix 445 Least squares line 361 Legendre polynomials 346 Local extremum 439 Local maximum 439 Local minimum 439 Lorentz transformation 454 Matrix representation of a bilinear form 424 Minimal solution of a system of equa-tions 364 Norm of a matrix 467 Norm of a vector 333 Normal matrix 370 Normal operator 370 Normalizing a vector 335 Orthogonal complement of a subset of an inner product space 349 Orthogonally equivalent, matrices 384 Orthogonal matrix 382 Orthogonal operator 379 Orthogonal projection 398 Orthogonal projection on a subspace 351 Orthogonal subset of an inner prod-uct space 335 Orthogonal vectors 335 Orthonormal basis 341 Orthonormal se;t. 335 Penrose conditions 421 Permanent, of a 2 x 2 matrix 448 Polar decomposition of a matrix 412 Pseudoinverse of a linear transforma-tion 413 Pseudoinverse of a matrix 414 Quadratic form 433 Rank of a bilinear form 443 Rayleigh quotient 467 Real inner product space 332 Reflection 473 Resolution of the identity operator induced by a linear transformation 402 Rigid motion 385 Rotation 472 Self-adjoint matrix 373 Self-adjoint operator 373 Signature of a form 444 Signature of a matrix 445 Singular value decomposition of a matrix 410 Singular value of a linear transforma-tion 407 Singular value of a matrix 410 Space-time coordinates 453 Spectral decomposition of a linear operator 402 Spectrum of a linear operator 402 Standard inner product 330 Symmetric bilinear form 428 Translation 386 Trigonometric polynomial 399 Unitarily equivalent matrices 384 Unitary matrix 382 Unitary operator 379 Unit vector 335 7 C a n o n i c a l F o r m s 7.1 7.2 7.3 7.4* The Jordan Canonical Form I The Jordan Canonical Form II The Minimal Polynomial The Rational Canonical Form . / I s we learned in Chapter 5, the advantage of a diagonalizable linear oper-ator lies in the simplicity of its elescription. Such an operator has a diagonal matrix representation, or, equivalently, there is an ordered basis for the un-derlying vector space consisting of cigenvee;tors of the operator. However, not every linear operator is diagonalizable, even if its characteristic polynomial splits. Example 3 of Section 5.2 describes such an operator. It is the purpose of this chapter to consider alternative matrix repre-sentations for nondiagonalizable operators. These representatiems are called canonical forms. There are different kinds of canonical forms, and their ad-vantages and disadvantages depend on how they are applied. The choice of a canonical form is determineel by the appropriate choice of an ordered basis. Naturally, the canonical forms of a linear ejperator are not diagonal matrices if the linear operator is not diagonalizable. In this chapter, we treat two common canonical forms. The first of these, the Jordan canonical form, requires that the characteristic polynomial of the operator splits. This form is always available if the underlying field is algebraically closed, that is, if every polynomial with coefficients from the field splits. For example, the field of complex numbers is algebraically closed by the fundamental theorem of algebra (see Appendix D). The first two sections deal with this form. The rational canonical form, treated in Section 7.4, does not require such a factorization. 7.1 THE JORDAN CANONICAL FORM I Let T be a linear operator on a finite-dimensional vector space V, and suppose that the characteristic polynomial of T splits. Recall from Section 5.2 that the eliagonalizability of T de;pends on whether the union of ordered bases for the distinct eigenspaces of T is an ordered basis for V. So a lack of diagonalizability means that at least one eigenspace of T is too 'small.' 482 Sec. 7.1 The Jordan Canonical Form I 483 In this section, we extend the definition of eigenspace to generalized eigenspace. From these subspaces, we select ordered bases whose union is an ordered basis (3 for V such that m * = /Ai O O A2 {O O 0 O Ak) where each O is a zero matrix, and each Ai is a square matrix of the form (A) or A 0 0 {0 1 A 0 0 0 1 0 0 0 0 A 0 > 0 1 V for some eigenvalue A of T. Such a matrix Ai is called a Jordan block corresponding to A, and the matrix [T]^ is called a Jordan canonical form of T. We also say that the ordered basis j3 is a Jordan canonical basis for T. Observe that each Jordan block Ai is 'almost' a diagonal matrixin fact, [T]^ is a diagonal matrix if and only if each Ai is of the form (A). Example 1 Suppose that T is a linear operator on C8, and (3 = {v,v2,... ,v&} is an ordered basis for C8 such that J = [T]/3 = ( 2 1 0 0 2 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 ^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 3 0 0 0 0 0 0 0 0 0 0 0 1 0 0 / is a Jordan canonical form of T. Notice that the characteristic polynomial of T is det(J ti) (t 2)4(t 3)2r.2, and hence the multiplicity of each eigenvalue is the number of times that the eigenvalue appears on the diagonal of J. Also observe that v,V4,v^, and vj are the only vectors in (3 that are eigenvectors of T. These are the vectors corresponding to the columns of J with no 1 above the diagonal entry. 484 Chap. 7 Canonical Forms J' = In Sections 7.1 and 7.2, we prove that every linear operator whose charac-teristic polynomial splits has a Jordan canonical form that is unique up to the order of the Jordan blocks. Nevertheless, it is not the case that the Jordan canonical form is completely determined by the characteristic polynomial of the operator. For example, let T' be the linear operator on C8 such that [T']/? = J ' , where (3 is the ordered basis in Example 1 and /2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0/ Then the characteristic polynomial of T' is also (t 2)4(t 3)2t2. But the operator T' has the Jordan canonical form J', which is different from J, the Jordan canonical form of the linear operator T of Example 1. Consider again the matrix J and the ordered basis (3 of Example 1. Notice that T(v2) = vi+2v2 and therefore, (T2)(v2) = v. Similarly, (T2)(v$) = v2. Since v and V4 are eigenvectors of T corresponding to A = 2, it follows that (T - 2)3(vi) = 0 for i = 1,2,3, and 4. Similarly (T - 3I)2(^) = 0 for i = 5,6, and (T - 0)2(vi) = 0 for i = 7,8. Because of the structure of each Jordan block in a Jordan canonical form, we can generalize these observations: If v lies in a Jordan canonical basis for a linear operator T and is associated with a Jordan block with diagonal entry A, then (T )p(v) 0 for sufficiently large p. Eigenvectors satisfy this condition for p = 1. Definition. Let T be a linear operator on a vector space V, and let A be a scalar. A nonzero vector x in V is called a generalized eigenvector of T corresponding to A if (T Al)p(x) = 0 for some positive integer p. Notice that if x is a generalized eigenvector of T corresponding to A, and p is the smallest positive integer for which (T Al)p(x) = 0, then (T Al)p_1(x) is an eigenvector of T corresponding to A. Therefore A is an eigenvalue of T. In the context of Example 1, each vector in (3 is a generalized eigenvector of T. In fact, vi,v2, V3 and V4 correspond to the scalar 2, i>5 and VQ correspond to the scalar 3, and V7 and vs correspond to the scalar 0. Just as eigenvectors lie in eigenspaces, generalized eigenvectors lie in 'gen-eralized eigenspaces.' Definition. Let T be a linear operator on a vector space V, and let A be an eigenvalue of T. The generalized eigenspace of T corresponding to Sec. 7.1 The Jordan Canonical Form I 485 A, denoted K, is the subset ofV defined by K = {x G V: (T Al)p(x) = 0 for some positive integer p}. Note that K consists of the zero vector and all generalized eigenvectors corresponding to A. Recall that a subspace W of V is T-invariant for a linear operator T if T(W) C W. In the development that follows, we assume the results of Exer-cises 3 and 4 of Section 5.4. In particular, for any polynomial g(x), if W is T-invariant, then it is also ^(T-invariant. Furthermore, the range of a linear operator T is T-invariant. Theorem 7.1. Let T be a linear operator on a vector space V, and let A be an eigenvalue of T. Then (a) K is a T-invariant subspace ofV containing E (the eigenspace ofT corresponding to X). (b) For any scalar p ^ A, the restriction ofT p to KA is one-to-one. Proof, (a) Clearly, 0 K. Suppose that x and y are in KA- Then there exist positive integers p and q such that Therefore (T-AI) p (x) = (T-AI)4(y) = 0. (T - X)P+fl(x + y) = (T- Al)p+(x) + (T - M)p+q(y) = (T-AI)(0)- r - (T-AI) p (0) = 0, and hence x + y G KA- The proof that KA is closed under scalar multiplication is straightforward. To show that K is T-invariant, consider any x G K. Choose a positive integer p such that (T Al)p(x) = 0. Then (T - AI)pT(x) = T(T - Al)p(x) = T(0) = 0. Therefore T(x) G K>. Finally, it is a simple observation that EA is contained in KA-(b) Let x G KA and (T p)(x) = 0. By way of contradiction, suppose that x 7^ 0. Let p be the smallest integer for which (T Al)p(x) = 0, and let y = (T-)P-l(x). Then (T-AI)(y) = (T-AI) p (x) = 0, and hence y G EA- Furthermore, (T - p)(y) = (T - p)(T - Wf-^x) = (T - AI)p-x(T - p)(x) = 0, so that j / 6 E,,. But EA n EM = {0}, and thus y = 0, contrary to the hypothesis. So x = 0, and the restriction of T p to KA is one-to-one. 1 486 Chap. 7 Canonical Forms Theorem 7.2. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits. Suppose that A is an eigenvalue of T with multiplicity m. Then (a) dim(KA) < m. (b) KA = N((T-AI) m ) . Proof, (a) Let W = KA, and let h(t) be the characteristic polynomial of T w By Theorem 5.21 (p. 314), h(t) divides the characteristic polynomial of T, and by Theorem 7.1(b), A is the only eigenvalue of Tw- Hence h(t) = (l)d(tX)d, where d = dim(W), and d < m. (b) Clearly N((T - Al)m) C KA. Now let W and h(t) be as in (a). Then /i(Tw) is identically zero by the Cayley-Hamilton theorem (p. 317); therefore (T - Al)d(x) = 0 for all x G W. Since d < m, we have KA C N((T - Al)m). | v Theorem 7.3. Let T be a linear operator on a finite-dimensional vec-tor~space V such that the characteristic polynomial of T splits, and let Ai, A2, , Afe be the distinct eigenvalues of T. Then, for every x G V, there exist vectors Vi G Kif 1 < i < k, such that X Vi + V2 H hUfe. Proof. The proof is by mathematical induction on the number k of dis-tinct eigenvalues of T. First suppose that k = 1, and let m be the multiplic-ity of Ai. Then (Ai t)m is the characteristic polynomial of T, and hence (Ail - T)m = To by the Cayley-Hamilton theorem (p. 317). Thus V = KAl, and the result follows. Now suppose that for some integer k > 1, the result is established when-ever T has fewer than k distinct eigenvalues, and suppose that T has k distinct eigenvalues. Let m be the multiplicity of Afc, and let f(t) be the characteristic polynomial of T. Then f(t) = (t Xk)mg(t) for some polynomial g(t) not divisible by (t - Xk). Let W = R((T - Afcl)m). Clearly W is T-invariant. Observe that (T Afcl)w maps KA{ onto itself for i < k. For suppose that i < k. Since (T Afcl)m maps Kt into itself and Afc 7^ Ai, the restriction of T Afc I to Ki is one-to-one (by Theorem 7.1(b)) and hence is onto. One consequence of this is that for i < k, K{ is contained in W, and hence Xi is an eigenvalue of Tw with corresponding generalized eigenspace Ki. Next, observe that Afc is not an eigenvalue of Tw- For suppose that T(v) = Afct? for some v G W. Then v = (T Afcl)m(y) for some y G V, and it follows that 0 = (T-Afcl)(t;) = (T-Afcir+ 1(2/)-Therefore y G KAfc. So by Theorem 7.2, v = (T - Xk)m(y) = 0. Since every eigenvalue of Tw is an eigenvalue of T, the distinct eigenvalues of T w are Ai, A 2 , . . . , Afc_i. Sec. 7.1 The Jordan Canonical Form I 487 Now let x G V. Then (T - Afcl)m(x) G W. Since T w has the k - 1 distinct eigenvalues Ai, A2, . . , Afc_i, the induction hypothesis applies. The corresponding generalized eigenspace of T ^ for each Xi is K., and hence there are vectors Wi G K^ 1 < i < k 1, such that (T - Afcl)m(x) = W! + w2 + + wk-i. Since (T Afcl)m maps KA; onto itself for i < k, there exist vectors Vi G KA; such that (T - Xk)m(vi) = Wi for i < k. Thus (T - Afcl)m(x) = (T - Xk)m(Vl) + (T - Afcl)m(t*) + + (T - Xk)m(vk-i), and it follows that x - (v + u2 H + vk-i) G KAfc. Therefore there exists a vector vk G KAfc such that x = v i + u 2 H +vk. | The next result extends Theorem 5.9(b) (p. 268) to all linear operators whose characteristic polynomials split. In this case, the eigenspaces are re-placed by generalized eigenspaces. Theorem 7.4. Let T be a linear operator on a finite-dimensional vec-tor space V such that the characteristic polynomial of T splits, and let Ai, A 2 , . . . , Afc be the distinct eigenvalues of T with corresponding multiplici-ties mi,m2,...,mk. For 1 < i < k, let Pi be an ordered basis for K{. Then the following statements are true. (a) A n f t = 0 fori^j. (b) (3 = 0i U (32 U U 0k is an ordered basis for V. (c) dim(KAj = mi for all i. Proof, (a) Suppose that x G /% fl f3j C KA4 D KAJ5 where i ^ j. By Theorem 7.1(b), T A^ l is one-to-one on KA^ , and therefore (T AJ)p(x) ^ 0 for any positive integer p. But this contradicts the fact that x G Kt, and the result follows. (b) Let x G V. By Theorem 7.3, for 1 < i < k, there exist vectors Vi G KA4 such that x = v + v2 + -f- vk. Since each Vi is a linear combination of the vectors of pi, it follows that x is a linear combination of the vectors of (3. Therefore (3 spans V. Let q be the number of vectors in (3. Then dim(V) < q. For each i, let di = dim(KAj). Then, by Theorem 7.2(a), q = >J di < T j m-i = dim(V). i=i i=i Hence q = dim(V). Consequently (3 is a basis for V by Corollary 2 to the replacement theorem (p. 47). 488 Chap. 7 Canonical Forms (c) Using the notation and result of (b), we se%e> that / J d t = > . ' m i - But i=l i di < m-i by Theorem 7.2(a). and the'reforc d, m, for all i. I Corollary. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits. Then T is diagonalizable if and only if EA = KA for every eigenvalue X ofT. Proof. Combining Theeuvms 7.1 and 5.0(a) (p. 268). we se>e> that T is diagonalizable' if anel only if (HIII(EA) = dim(KA) for each eigenvalue A of T. But EA G KA, anel he'iure; the>se; subspnce>s have1 the same dimension if and only if they are equal. We now foe-us our attention on the' problem of select ing suitable bases for the generalized eigenspaces of a linear operates se> that we- may use' The'o-rem 7.4 to obtain a Jordan canonical basis for the operator. For this purpose;, we consieler again the; basis ft of Example 1. We; have seen that the first four vectors of (3 lie' in the generalize'el eigenspace K2. Observe that the vee-te)rs in 0 that eletcrinine the first Jorelan block of ./ are of the; form {'i.'2.':i} = { (T-2 l ) 2 ( e 3 ) . (T -2 l ) ( r0 . r : { } . Furthermore, e)bse>rve that ( T - 2l)^ (/.{) = 0. The relation be;twe;e;n these vcc-te>rs is the key to finding Jordan canonie-al base's. This leads to the fe)lle)wing ele'finitions. Definitions. Let T be a linear operator on a vector space V. and let x be a generalized eigenvector of T corresponding to the eigenvalue X. Suppose that p is the smallest positive integer for which (T - Al)p(x) = 0. Then the ordered set {(T - AI)'1 (x). (T - Al)'-2(x), . . . , (T - Al)(x),x} is called a cycle of generalized eigenvectors of T corresponding to X. The vectors (T Al)p~'(x) and x are called the initial vector and the end vector of the cycle, respectively. We say that the length of the cycle is p. Notice' that the initial vector of a cycle of generalized eigenvecte>rs of a line;ar operator T is the only eigenvector erf T in the eyerie1. Alse) observe that if.r is an eigenvector of T e-orrosponding to the eigenvalue A, then the set {x} is a cycle of generalized eigenvectors of T corresponding to A of length 1. In Example 1, the subsets fti {v,v2,v;}, 02 = {'4}^ 03 = {?>Sec. 7.1 The Jordan Canonical Form I 489 Theorem 7.5. Let T be a linear operator on a finite-dimensional vector space V whose characteristic polynomial splits, and suppose that ft is a basis for V such that ft is a disjoint union of cycles of generalized eigenvectors of T. Then the following statements are true. (a) For each cycle 7 of generalized eigenvectors contained in ft, W = span(7) is T-invariant, and [Tw]7 is a Jordan block. (b) 0 is a Jordan canonical basis for V. Proof, (a) Suppose that 7 corresponds to A, 7 has length p, and x is the end vector of 7. Then 7 = {v, w2 , . . . , vp}, where Vi = (T Al)p-*(x) for i < p and vp x. So ( T - A I ) M = (T-Al ) p (x ) = 0, and hence T(v{) = Xv. For i > 1, ( T - A I ) K ) = ( T - A I ) ^ i - 1 > ( x ) = ^ _ i . Therefore T maps W into itself, and, by the preceding equations, we sec that [Twh is a Jorelan block. For (b), simply repeat the arguments of (a) for each cycle in ft in order to obtain [T]^. We leave the details as an exercise;. II In view of this result, we must show that, under appropriate conditions, there exist bases that are disjoint unions of cycles of generalized eigenvectors. Since the characteristic polynomial of a Jordan canonical form splits, this is a neec;ssary condition. We will soon s490 Chap. 7 Canonical Forms For each i, let 7^ denote the cycle obtained from 7$ by deleting the end vector. Note that if 7* has length one, then 7^ = 0 . In the case that 7^ ^ 0 , each vector of 7^ is the image under U of a vector in 7;, and conversely, every nonzero image under U of a vector of 7$ is contained in 7^. Let 7' = IIT*-i Then by the last statement, 7' generates R(U). Furthermore, 7' consists of n q vectors, and the initial vectors of the 7^'s are also initial vectors of the 7i's. Thus we may apply the induction hypothesis to conclude that 7' is linearly independent. Therefore 7' is a basis for R(U). Hence dim(R(U)) = n q. Since the q initial vectors of the 7,;'s form a linearly independent set and lie in N(U), we have dim(N(U)) > q. From these inequalities and the dimension theorem, we obtain n > dim(W) = dim(R(U)) ->(n-q) + q n. dim(N(U)) We conclude that dim(W) = n. Since 7 generates W and consists of n vectors, it must be a basis for W. Hence 7 is linearly independent. I Corollary. Every cycle of generalized eigenvectors of a linear operator is linearly independent. Theorem 7.7. Let T be a linear operator on a finite-dimensional vector space V, and let X be an eigenvalue of T. Then K has an ordered basis con-sisting of a union of disjoint cycles of generalized eigenvectors corresponding to X. Proof. The proof is by mathematical induction on n = dim(KA). The result is clear for n = 1. So suppose that for some integer n > 1 the result is valid whenever dim(KA) < n, and assume that dim(KA) = n. Let U denote the restriction of T AI to KA- Then R(U) is a subspace of KA of lesser dimension, and R(U) is the space of generalized eigenvectors corresponding to A for the restriction of T to R(U). Therefore, by the induction hypothesis, there exist disjoint cycles 71,72 , . . , 7g of generalized eigenvectors of this restriction, and 1 hence of T itself, corresponding to A for which 7 = M 7^ is a basis for R(U). For 1 < i < q, the end vector of 7 is the image under U of a vector Vi G KA, and so we can extend each 7* to a larger cycle ji = 7* U {vi} of generalized eigenvectors of T corresponding to A. For 1 < i < q, let Wi be the initial vector of 7J (and hence of 7*). Since {wi,W2, , wq} is a linearly independent sub-set of EA, this set can be extended to a basis {wi, w 2 , . . . ,wQ,U,U2,... ,us} Sec. 7.1 The Jordan Canonical Form I 491 for EA- Then 71,72, , % , {i}, {^2}, , {'} are disjoint cycles of gener-alized eigenvectors of T corresponding to A such that the initial vectors of these cycles are linearly independent. Therefore their union 7 is a linearly independent subset of KA by Theorem 7.6. We show that 7 is a basis for KA- Suppose that 7 consists of r = rank(U) vectors. Then 7 consists of r + q + s vectors. Furthermore, since {wi,W2, ,wq, U, w 2 , . . . , us] is a basis for EA = N(U), it follows that nullity(U) = q + s. Therefore dim(KA) = rank(U) + nullity(U) = r + q + s. So 7 is a linearly independent subset of KA containing dim(KA) vectors. It follows that 7 is a basis for KA- I The following corollary is immediate. Corollary 1. Let T be a linear operator on a finite-dimensional vec-tor space V whose characteristic polynomial splits. Then T has a Jordan canonical form. Proof. Let Ai, A 2 , . . . . Afc be the distinct eigenvalues of T. By Theorem 7.7, for each i there is an ordered basis fti consisting of a disjoint union of cycles of generalized eigenvectors corresponding to A,. Let ft = ft U 02 U U 0k-Then, by Theorem 7.4(b), ft is an ordered basis for V. 1 The Jordan canonical form also can be studied from the viewpoint of matrices. Definition. Let A G Mn X n(F) be such that the characteristic polynomial of A (and hence of LA) splits. Then the Jordan canonical form of A is defined to be the Jordan canonical form of the linear operator LA on Fn. The next result is an immediate consequence of this definition and Corol-lary 1. Corollary 2. Let A be an n x n matrix whose characteristic polynomial splits. Then A has a Jordan canonical form J, and A is similar to J. Proof. Exercise. 1 We can now compute the Jordan canonical forms of matrices and linear operators in some simple cases, as is illustrated in the next two examples. The tools necessary for computing the Jordan canonical forms in general are developed in the next section. 492 Example 2 Let Chap. 7 Canonical Forms A= -3 1 1 1 0 - 1 - 2 5 4 G M3x3(rt). To find the Jorelan canonical form for A, we; need to find a Jordan canonical basis for T = L.,. The characteristic polynomial of A is f(t)=det(A-tI) = -(t-Z)(t-2)2. Hence Ai = 3 and A2 = 2 arc the eigenvalues of A with multiplicities 1 and 2, respectively. By Theorem 7.4, dim(KA,) = 1, and dhn(KA2) = 2. By Theorem 7.2. KA, = N(T-3I), and KA, = N((T-2I)2). Since EAl = N(T-3I), we have that EA, = KA, . Observe that (1,2,1) is an eigenvector of T corresponding to A) = 3; therefore ft = is a basis for KA, . Since dim(KA_,) = 2 and a generalized eigenspace has a basis consisting of a union of cycles, this basis is either a union of two cycles of length 1 or a single cycle of length 2. The former case is impossible because the vectors in the basis would be eigenvectors -contradicting the fact that dim(EA2) = 1. Therefore the desired basis is a single cycle of length 2. A vector v is the end vector of such a cycle if and only if (A - 2I)v f 0, but (A - 2I)2v = 0. It can easily be shown that is a basis for the solution space of the homogeneous system (A 2I)2.v = 0. Now choose a vector v in this set so that (A 2I)v ^ 0. The vector v = (-1,2,0) is an ae-c;e;ptable candidate for v. Since (A 2I)v = (I, 3, 1), we obtain the cycle of generalizes! eigenvectors 02 = {(A-2I)v,v} = m Sec. 7.1 The Jordan Canonical Form I 493 as a basis for KA2 Finally, we take the union of these two bases to obtain ft = ftiUft2 = which is a Jordan canonical basis for A. Therefore, J = m * = is a Jordan canonical form for A. Notice that A is similar to J. In fact, J = Q~lAQ, where Q is the matrix whose columns are the vectors in 0. 3 ~o1 0 0 2 0 0 1 2 Example 3 Let T be the linear operator on P2(R) defined by T(g(x)) = g(x) g'(x). We find a Jordan canonical form of T and a Jordan canonical basis for T. Let ft be the standard ordered basis for P2(R)- Then m * = which has the characteristic polynomial f(t) (t + l)3 . Thus A = 1 is the only eigenvalue of T, and hence KA = ?2(R) by Theorem 7.4. So ft is a basis for KA- NOW dim(EA) = 3 - rank(j4 + 7) = 3 - rank '0 - 1 0 0 0 - 2 = 3 - 2 = 1. .0 0 0 / Therefore a basis for KA cannot be a union of two or three cycles because the initial vector of each cycle is an eigenvector, and there do not exist two or more linearly independent eigenvectors. So the desired basis must consist of a single cycle of length 3. If 7 is such a cycle, then 7 determines a single Jordan block / - I 1 0' [T], = 0 - 1 1 0 0 - 1 which is a Jordan canonical form of T. The end vector h(x) of such a. cycle must satisfy (T 4- )2(h(x)) / 0. In any basis for KA, there must be a vector that satisfies this condition, or else 494 Chap. 7 Canonical Forms no vector in KA satisfies this condition, contrary to our reasoning. Testing the vectors in 0, we see that h(x) = x is acceptable. Therefore 7 = {(T + I ) V ) , ( T + I)(x2),x2} = {2,~2x,,;2} is a Jordan canonical basis for T. In the next section, we develop a computational approach for finding a Jordan canonical form and a Jordan canonical basis. In the process, we; prove that Jorelan canonical forms are unique up to the order of the Jordan blocks. Let T be a linear operator on a finite-dimensional vector space V, and sup-pose that the characteristic polynomial of T splits. By Theorem 5.11 (p. 278), T is diagonalizable if and only if V is the direct sum of the eigenspaces of T. If T is diagonalizable. then the eigenspaces and the generalized eigenspaces coincide. The next result, which is optional, extends Theorem 5.11 to the nondiagonalizable case. Theorem 7.8. Let T be a linear operator on a finite-dimensional vector space V whose characteristic polynomial sj>lits. Then V is the direct sum of the generalized eigenspaces of T. Proof. Exercise. ( EXERCISES 1. Label the following statements as true or false. (a) Eigenvectors of a linear operator T are also generalized eigenvec-tors of T. (b) It is possible; for a generalized eigenvector of a linear operator T to correspond to a scalar that is not an eigenvalue of T. (c) Any linear operator on a finite-dimensional ve;e:tor space; has a Jor-elan canonical form. (d) A cycle of generalized eigenvectors is linearly independent. (e) There is exactly one cycle of generalized eigenvectors correspond-ing to each eigenvalue of a linear operator on a finite-dimensional ve;ctor space;. (f) Let T be a linear operator on a finite-dimensional vector space whose characteristic polynomial splits, and let Ai,A2 Afc be the; distinct eigenvalues of T. If, for each i, fti is a basis for K,,, then 0i U 02 U U ftk is a Jordan canonical basis for T. (g) For any Jorelan block ./, the operator L./ has Jordan canonical form J. (h) Let T be a linear operator on an n-dimensional vector spae;e; whose; characteristic polynomial splits. Then, for any eigenvalue A of T, KA = N((T-AI)') . Sec. 7.1 The Jordan Canonical Form I 495 2. For each matrix A, find a basis for each generalized eigenspace of L^ consisting of a union of disjoint cycles of generalized eigenvectors. Then find a Jordan canonical form J of A. (a) A = (c) A = 1 - 1 11 21 3 l *) - 4 - 8 - 1 - 5 - 1 1 0 (b) A = (d) A = 1 2 3 2 (2 1 0 2 0 0 0 o 1 0 3 0 ^0 1 - 1 3/ 3. For each linear operator T, find a basis for each generalized eigenspace of T consisting of a union of disjoint cycles of generalized eigenvectors. Then find a Jordan canonical form J of T. (a) T is the linear operator on P2(R) defined by T(/(x)) = 2/(x) (b) V is the real vector space of functions spanned by the set of real valued functions {l,t, t2,e1, te1}, and T is the linear operator on V defined by T( / ) = / ' . (c) T is the linear operator on M 2 x 2 (#) defined by T(A) = I ) -A for all A G M2X2(#). (d) T(A) = 2A + At for all A G M2x2(i2). 4 J Let T be a linear operator on a vector space V, and let 7 be a cycle of generalized eigenvectors that corresponds to the eigenvalue A. Prove that span(7) is a T-invariant subspace of V. 5. Let 71,72, ,7P be cycles of generalized eigenvectors of a linear op-erator T corresponding to an eigenvalue A. Prove that if the initial eigenvectors are distinct, then the cycles are disjoint. 6. Let T: V - W b e a linear transformation. Prove the following results. (a) N(T) = N(-T). (b) N(Tfc) = N((-T) fc). (c) If V = W (so that T is a linear operator on V) and A is an eigen-value of T, then for any positive integer k N((T-Alv) f c) = N((Alv-T) f c). 7. Let U be a linear operator on a finite-dimensional vector space V. Prove the following results. (a) N(U) C N(U2) C C N(Ufc) C N(Ufc+1) C . 496 Chap. 7 Canonical Forms 8. 9. 10. 11. (b) (c) (d) (e) (f) If rank(Um) = rank(Um+1) for some positive integer rn, then rank(Um) = rank(Ufc) for any positive integer k > m. If rank(UTO) = rank(Uw+1) for some positive integer rn, then N(U') = N(Ufc) for any positive integer k > rn. Let T be a linear operator on V. and let A be an eigenvalue of T. Prove that if rank((T-Al)'n) = rank((T-Al)m + 1) for some integer rn, then KA = N((T - AI)')-Second Test for Diagonalizability. Let T be a linear operator on V whose characteristic polynomial splits, and let Ai, A2 , . . . , Afc be the distinct eigenvalues of T. Then T is diagonalizable if and only if rank(T - A;I) = rank((T - A,I)2) for 1 < i < k. Use (e) to obtain a simpler proof of Exercise 24 of Section 5.4: If T is a diagonalizable linear operator on a finite-dimensional vec-tor space V and W is a T-invariant subspace of V. then Tw is diagonalizable. Use Theorem 7.4 to prove that the vectors v, v2,..., vk in the statement of Theorem 7.3 are unique. Let T be a linear operator on a finite-dimensional vector space V whose characteristic polynomial splits. (a) Prove Theorem 7.5(b). (b) Suppose; that ft is a Jordan canonical basis for T. and let A be an eigenvalue of T. Let ft' = ft n KA- Prove that ft' is a basis for KA-Let T be a linear operator on a finite-dimensional vector space whose characteristic polynomial splits, and let A be an eigenvalue of T. (a) Suppose that 7 is a basis for KA consist ing of the union of q disjoint cycles of generalized eigenvectors. Prove that q < dim(EA). (b) Let ft be a Jordan canonical basis for T, and suppose that J = [T]^ has q Jordan blocks with A in the diagonal positions. Prove that q < dim(EA). Prove Corollary 2 to Theorem 7.7. Exercises 12 and 13 are concerned with direct sums of matrices, defined in Section 5.4 on page 320. 12. Prove Theorem 7.8. 13. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let Ai, A 2 , . . . , Afc be the distinct eigenvalues of T. For each i, let Jt be the Jordan canonical form of the restriction of T to KA^ . Prove that J - J J 2 Jfc is the Jordan canonical form of J . Sec. 7.2 The Jordan Canonical Form II 7.2 THE JORDAN CANONICAL FORM II 497 For the purposes of this section, we fix a linear operator T on an n-dimensional vector space V such that the characteristic polynomial of T splits. Let Ai. A2 Afc be the distinct eigenvalues of T. By Theorem 7.7 (p. 490), each generalized eigenspace KA, contains an ordered basis fti consisting of a union of disjoint cycles of generalized eigen-vectors corresponding to A,. So by Theorems 7.4(b) (p. 487) and 7.5 (p. 489). fc the union ft M fti is a Jordan canonical basis for T. For each i, let T, be the restriction of T to KA,, anel let Ai = [T,],^. Then Ai is the Jordan canonical form of T,. and J = m * = /Ai O 0 A2 () O 0 0 Ak) is the Jordan canonical form of T. In this matrix, each O is a zero matrix of appropriate size. In this section, wc compute the matrices A, and the bases fti, thereby computing ./ anel ft as well. While developing a method for finding ,/, it becomes evident that in some sense the matrices Aj are unique. To aid in formulating the uniqueness theorem for ./, we adopt the following convention: The basis fti for KA, will henceforth be ordered in such a way that the cycles appear in order of decreasing length. That is, if/?,; is a disjoint union of cycles 7!, 72 , . . . . 7,,, and if the length of the cycle 7~ is pj, we index the ewcles so that p > p2 > > pUi. This ordering of the cye:les limits the possible; orderings of vectors in fti, which in turn determines the matrix Aj. It is in this sense that A, is unique. It then follows that the Jordan canonical form for T is unique up to an ordering of the eigenvalues of T. As we will see, there; is no uniqueness theorem for the; bases fti or for ft. Specifically, we show that for eae:h i. the number n, of cycles that form 0i} and the length pj (j 1 .2. . . . , rij) of e%ach cycle, is completely detenninc'd by T. Example 1 To illustrate the discussion above, suppose that, for some /', the ordered basis fti for KA, is the union of four cycles fti = 71 U 72 U 73 U 74 with respective 498 Chap. 7 Canonical Forms lengths p] = 3, p2 = 3, pz = 2, and p4 = 1. Then Ar = (Xi 0 0 0 0 0 0 0 1 A, 0 0 0 0 0 0 0 0 1 A, 0 0 0 0 0 0 0 0 0 A, 0 0 0 0 0 0 0 0 1 A, 0 0 0 0 0 0 0 0 1 A, 0 0 0 0 0 0 0 0 0 Xi 0 0 0 0 0 0 0 0 1 A, 0 0 0 0 0 0 0 0 0 Xi ) To help us visualize each of the matrices A, anel ordered bases 0i, we use an array of dots called a dot d iagram of T;. where T, is the restriction of T to KA,. Suppose that fti is a disjoint union of cycles of generalized eigenvectors 71,72, 7n< with lengths p > p2 > > pn., respectively. The dot diagram of T,; contains one clot for each vector in fti, anel the dots are configured according to the following rules. 1. The array consists of n* columns (one column for each cycle). 2. Counting from left to right, the y'th column consists of the Pj dots that correspond to the vectors of 7/ starting with the initial vector at the top and continuing down to the end vector. Denote the end vectors of the cycles by v.v2 2; (T-A, l ) ( 2 ) v2 ( T - A t l ) p . - ' K , ) (T - iri-2(vni) (T-K)(vni) ( T - A , l ) ( c 1 ) e, Notice that the; dot diagram of Tj has //, columns (one for each cycle;) and p rows. Since; pi > p2 > > pni- the columns of the dot diagram become shorter (or at least not longer) as we move from left to right. Now let /-, denote the number of dots in the j t h row of the dot diagram. Observe that r > r2 > > r P l . Furthermore, the diagram can be re-constructed from the values of the r'j's. The proofs of these; facts, which are combinatorial in nature, are treated in Exercise 9. Sec. 7.2 The Jordan Canonical Form II 499 In Example 1, with n-i = 4, p = p2 = 3, p$ = 2, and P4 = 1, the dot diagram of Tj is as follows: Here r = 4, r2 = 3, and r3 = 2. We now devise a method for computing the dot diagram of Tj using the ranks of linear operators determined by T and A .^ Hence the dot diagram is completely determined by T, from which it follows that it is unique. On the other hand, 0i is not unique. For example, see Exercise 8. (It is for this reason that we associate the dot diagram with T,; rather than with fti.) To determine the dot diagram of Ti, we devise a method for computing each Tj, the number of dots in the j th row of the dot diagram, using only T and Xi. The next three results give us the required method. To facilitate our arguments, we fix a basis 0i for KA; SO that fti is a disjoint union of n7; cycles of generalized eigenvectors with lengths p > p2 > > prli. Theorem 7.9. For any positive integer r, the vectors in fti that arc associated with the dots in the first r rows of the dot diagram of Ti constitute a basis for N((T AJ)r). Hence the number of dots in the first r rows of the dot diagram equals nullity((T AJ)7-). Proof. Clearly, N((T-A t l ) r ) C KA,;, and KXi is invariant under (T-A ?J) r . Let U denote the restriction of (T Ajl)7' to KA*. By the preceding remarks, N((T Ajl)r) = N(U), and hence it suffices to establish the theorem for U. Now define Si = {x G fti: U(x) = 0} and S2 = {./; G A : U(x) ^ 0}. Let a and b denote the number of vectors in S and S2, respectively, and let m-i = din^KAj. Then a + b = rrii. For any x G fti, x G Si if and only if x is one of the first r vectors of a cycle, and this is true if and only if x corresponds to a dot in the first r rows of the dot diagram. Hence a is the number of dots in the first r rows of the dot diagram. For any x & S2, the effect of applying U to x is to move the dot corresponding to x exactly r places up its column to another dot. It follows that U maps 52 in a one-to-one fashion into fti. Thus {U(x): x G S2} is a basis for R(U) consisting of b vectors. Hence rank(U) = b, and so nullity(U) = nii b = a. But S is a linearly independent subset of N(U) consisting of a vectors; therefore 5] is a basis for N(U). 1 In the case that r = 1, Theorem 7.9 yields the following corollary. Corollary. The dimension of EA; is n. Hence in a Jordan canonical form ofT, the number of Jordan blocks corresponding to Xi equals the dimension of Ex,-500 Chap. 7 Canonical Forms Proof. Exercise. We are now able to devise a method for describing the dot diagram in terms of the ranks of operators. Theorem 7.10. Let rj denote the number of dots in the jth row of the dot diagram of Tj, the restriction ofT to KA4. Then the following statements are true. (a) n = dim(V) - rank(T - AJ). (b) rj = rank((T - AJ) ' - 1 ) - rank((T - Xi)j) if j > 1. Proof. By Theorem 7.9, for 1 < j < pi, we have ri + r2 + + rj = nullity((T - Ajl)j) = dim(V) - rank((T - Ail)J). Hence r dim(V) rank(T Al), and for j > 1, rj = (rL +r2- + Tj) - ( n + r2 + h r^- i) = [dim(V) - rank((T - Xi)j)} - [dim(V) - rank((T - Ajl)^1)] = rank((T - Ajl)^1) - rank((T - Xi)j). I Theorem 7.10 shows that the dot diagram of Tj is completely determined by T and Aj. Hence we have proved the following result. Corollary. For any eigenvalue Xi ofT, the dot diagram of Ti is unique. Thus, subject to the convention that the cycles of generalized eigenvectors for the bases of each generalized eigenspace are listed in order of decreasing length, the Jordan canonical form of a linear operator or a matrix is unique up to the ordering of the eigenvalues. We apply these results to find the Jordan canonical forms of two matrices and a linear operator. Example 2 Let A = (1 0 0 V0 - l 3 1 - 1 0 1 - 1 0 1 0 0 3 / Sec. 7.2 The Jordan Canonical Form II 501 We find the Jordan canonical form of A and a Jordan canonical basis for the linear operator T = L^. The characteristic polynomial of A is det(i4 - ti) = (t- 2)3(i - 3). Thus A has two distinct eigenvalues, Ai = 2 and A2 = 3, with multiplicities 3 and 1, respectively. Let Ti and T2 be the restrictions of L^ to the generalized eigenspaces KA, and KA2, respectively. Suppose that 0 is a Jordan canonical basis for Ti. Since Ai has multi-plicity 3, it follows that dim^A,) = 3 by Theorem 7.4(c) (p. 487); hence the dot diagram of Ti has three dots. As we did earlier, let rj denote the number of dots in the jth. row of this dot diagram. Then, by Theorem 7.10, n = 4 rank(i4 27) = 4 rank / o 0 0 V> - 1 1 1 - 1 0 - 1 - 1 0 i 0 0 V = 4 - 2 = 2, and r 2 = ram(A - 21) - rank((,4 - 21 f) = 2 - 1 = 1. (Actually, the computation of r2 is unnecessary in this case because r = 2 and the dot diagram only contains three dots.) Hence the dot diagram associated with 0i is So (2 1 0' M = P'ilA = 0 2 0 V 0 2, Since A2 = 3 has multiplicity 1, it follows that dim(KA2) = 1, and conse-quently any basis 02 for KA2 consists of a single eigenvector corresponding to A2 = 3. Therefore A2 = [T2J>a = (3). Setting 0 = 0 U 02, we have J=[lAb = (2 1 0 0 0 2 0 0 0 0 2 0 0 0 0 3 / 502 Chap. 7 Canonical Forms and so J is the Jordan canonical form of A. We now find a Jordan canonical basis for T = L^. We begin by determin-ing a Jordan canonical basis 0 for Ti- Since the dot diagram of Ti has two columns, each corresponding to a cycle of generalized eigenvectors, there are two such cycles. Let v and u2 denote the end vectors of the first and second cycles, respectively. We reprint below the dot diagram with the dots labeled with the names of the vectors to which they correspond. (T-2l)(t>i) v2 Prom this diagram we sec that V e N((T - 2I)2) but Vi N(T - 21). Now / 0 - 1 0 1 0 1 0 - 1 It is easily seen that A-2J = 0 - 1 - 1 0 1 0 0 and (A - 2I)2 = /() 0 0 V' - 2 0 0 - 2 0 0 f ( 0 0 W ' AA l 2 w / o I 0 w is a basis for N((T 2I)2) = KA, . Of these three basis vectors, the last two do not belong to N(T 21), and hence we select one of these for v. Suppose that we choose 'i = Then (0 1 2 w /o - i 0 1 0 1 V --1 0 - 1 - 1 0 (T-2)(vi) = (A-2I)(v1) = Now simply choose v2 to be a vector in EA, that is linearly independent of (T - 21) (t>i); for example, select L 0 0 V (0 1 2 w -* -1 ' -1 V - i / V2 = / 1 0 0 Sec. 7.2 The Jordan Canonical Form II 503 Thus we have associated the Jordan canonical basis 0i = with the dot diagram in the following manner. M - 1 - 1 K-y > ( i 2 sPJ ) (1 0 0 v504 Chap. 7 Canonical Forms Example 3 Let ( 2 -2 -2 V-2 - 4 0 - 2 - 6 2 1 3 3 2^ 3 3 V A = We find the Jordan canonical form J of A, a Jordan canonical basis for L^, and a matrix Q such that J Q _ 1 AQ. The characteristic polynomial of A is dct(A - ti) = (t - 2)2(t - 4)2. Let T = L^, Ai = 2, and A2 = 4, and let T; be the restriction of L^ to KA* for i = 1,2. We begin by computing the dot diagram of Ti. Let r denote the number of dots in the first row of this diagram. Then r, = 4 - rank(,4 - 21) = 4 - 2 = 2; hence the dot diagram of Ti is as follows. Therefore ^ = [ T , ] , , = g ) where /?i is any basis corresponding to the dots. In this case, 0 is an arbitrary basis for EAX = N(T 21), for example, 01 = Next we compute the dot diagram of T2. Since rank(.A 4/) = 3, there is only 4 3 = 1 dot in the first row of the diagram. Since A2 = 4 has multiplicity 2, we have dim(KA2) = 2, and hence this dot diagram has the following form: / 2 1 0 v 5 M 1 2 w Thus M = [T2]A = 4 1 0 4 Sec. 7.2 The Jordan Canonical Form II 505 where 02 is any basis for KA., corresponding to the dots. In this case, 02 is a cycle of length 2. The end vector of this cycle is a vector v K^a = N((T - 4I)2) such that v N(T - 41). One way of finding such a vector was used to select the vector v in Example 2. In this example, we illustrate another method. A simple calculation shows that a basis for the null space of L^ - 41 is Choose v to be any solution to the system of linear equations / 0 {A - AI)x = 1 W for example, Thus - 1 - 1 V 02 = {(LA-4)(v),v} = { ( / 0 1 1 I W 1 l - i - i V o / Therefore (2y /& M 1 o 2 / 1 2 < W 1 1 v) - I - 1 V 506 Chap. 7 Canonical Forms f2 1 0 V2 0 0 1 1 2 1 0 1 1 - 1 - 1 V Finally, we define Q to be the matrix whose columns are the vectors of 0 listed in the same order, namely, Q = Then J = Q~lAQ. Example 4 Let V be the vector space of polynomial functions in two real variables x and y of degree at most 2. Then V is a vector space over R and a = {l,x,y, x2,y2,xy} is an ordered basis for V. Let T be the linear operator on V defined by T(f(x,y)) = -^f(x,y). For example, if f(x, y) = x + 2x2 3xy + y, then TCffoy)) = j- (x + 2x2 - 3xy + y) = 1 + Ax - 3y. We find the Jordan canonical form and a Jordan canonical basis for T. Let A = [T]a. Then /() 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 vo o o o o oy A = and hence the characteristic polynomial of T is det(A - ti) = det f-t 0 0 0 0 V () I -t 0 0 0 0 0 0 -t 0 0 0 0 2 0 -t 0 0 0 0 0 0 -t 0 0 0 1 0 0 - v = ? Thus A = 0 is the only eigenvalue of T, and KA = V. For each j, let rj denote the number of dots in the jib. row of the dot diagram of T. By Theorem 7.10, n = 6 - rank(;4) = 6 - 3 = 3, Sec. 7.2 The Jordan Canonical Form II 507 and since .42 = A) 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / r2 = rank(4) - rank(.42) = 3 - 1 = 2. Because there are a total of six dots in the dot diagram and r = 3 and r2 = 2, it follows that r$ = 1. So the dot diagram of T is We conclude that the Jordan canonical form of T is J = / 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 o 0 0 0 0 0 / We now find a Jordan canonical basis for T. Since the first column of the dot diagram of T consists of three dots, we must find a polynomial f(x,y) d2 :fi(x,y) 7^ 0. Examining the basis a = {l,x,y,x2,y2,xy} for such that dx2 KA = V, we see that x is a suitable candidate. Setting fi(x,y) = x , we see that (T - X)(h(x,y)) = T(h(x,y)) = (x2) = 2x and (T - X)2(j(x,y)) = T2(h(x,y)) = ^(x2) = 2. Likewise, since the second column of the dot diagram consists of two dots, we must find a polynomial /2(a;, y) such that ~(f2(x,y))^0, but ^(f2(x,y)) = 0. 508 Chap. 7 Canonical Forms Since our choice must be linearly independent of the polynomials already chosen for the first cycle, the only choice in a that satisfies these constraints is xy. So we set /2(a;.?y) = xy. Thus (T - X)(f2(x>y)) = T(f2(x,y)) = %-(xy) = y. ox Finally, the third column of the dot diagram consists of a single polynomial that lies in the null space of T. The only remaining polynomial in a is y2, and it is suitable here. So set fz(x,y) = y2. Therefore we have identified polynomials with the dots in the dot diagram as follows. 2 2x x2 y >xy ir Thus 0 = {2,2x,x2,y, xy,y2} is a Jordan canonical basis for T. In the three preceding examples, we relied on our ingenuity and the con-text of the problem to find Jordan canonical bases. The reader can do the same in the exercises. We are successful in these cases because the dimen-sions of the generalized eigenspaces under consideration are small. We do not attempt, however, to develop a general algorithm for computing Jordan canonical bases, although one could be devised by following the steps in the proof of the existence of such a basis (Theorem 7.7 p. 490). The following result may be thought of as a corollary to Theorem 7.10. Theorem 7.11. Lot A and B he n x n matrices, each having Jordan canonical forms computed according to the conventions of this section. Then A and B are similar if and only if they have (up to an ordering of their eigenvalues) the same Jordan canonical form. Proof. If A and B have the same Jordan canonical form J, then A and B are each similar to J and hence are similar to each other. Conversely, suppose that A and B are similar. Then A and B have the same eigenvalues. Let J A and JR denote the Jordan canonical forms of A and B, respectively, with the same ordering of their eigenvalues. Then A is similar to both J A and JB, and therefore, by the corollary to Theorem 2.23 (p. 115), J A and JB are matrix representations of L^. Hence J A and JB are Jordan canonical forms of L^. Thus J A = JB by the corollary to Theorem 7.10. 1 Example 5 We determine which of the matrices B = Sec. 7.2 The Jordan Canonical Form II 509 '0 1 2' and /; [ o i l .0 0 2, are similar. Observe that A, B, and C have the same characteristic poly-nomial (t l)(t 2)2, whereas D has t(t l)(t 2) as its characteristic polynomial. Because similar matrices have the same characteristic polynomi-als, D cannot be similar to A, B, or C. Let J A, JB, a n d Jc be the Jordan canonical forms of A, B, and C, respectively, using the ordering 1, 2 for their common eigenvalues. Then (see Exercise 4) JB = Since J A = Jc-, A is similar to C. Since JB is different from J A and Jc, B is similar to neither A nor C. The reader should observe that any diagonal matrix is a Jordan canonical form. Thus a linear operator T on a finite-dimensional vector space V is diag-onalizable if and only if its Jordan canonical form is a diagonal matrix. Hence T is diagonalizable if and only if the Jordan canonical basis for T consists of eigenvectors of T. Similar statements can be made about matrices. Thus, of the matrices A, B, and C in Example 5, A and C are not diagonalizable because their Jordan canonical forms are not diagonal matrices. 0 0 0 2 0 o , 2 and Jc = / I Vo 0 2 0 0 1 2 EXERCISES 1. Label the following statements as true or false. Assume that the char-acteristic polynomial of the matrix or linear operator splits. (a) The Jordan canonical form of a diagonal matrix is the matrix itself. (b) Let T be a linear operator on a finite-dimensional vector space V that has a Jordan canonical form J. If 0 is any basis for V, then the Jordan canonical form of [T]^ is J . (c) Linear operators having the same characteristic polynomial are similar. (d) Matrices having the same Jordan canonical form are similar. (e) Every matrix is similar to its Jordan canonical form. (f) Every linear operator with the characteristic polynomial (l)n(t A)ra has the same Jordan canonical form. (g) Every linear operator on a finite-dimensional vector space has a unique Jordan canonical basis. (h) The dot diagrams of a linear operator on a finite-dimensional vec-tor space are unique. 510 Chap. 7 Canonical Forms 2. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits. Suppose that Ai = 2, A2 = 4, and A3 = 3 are the distinct eigenvalues of T and that the dot diagrams for the restriction of T to KA, (i = 1,2,3) are as follows: Ai = 2 A, = 4 A3 = - 3 Find the Jordan canonical form J of T. 3. Let T be a linear operator on a finite-dimensional vector space V with Jordan canonical form / 2 1 0 0 2 1 0 0 2 0 0 0 0 0 0 0 0 0 V 0 0 0 0 0 0 0 N 0 0 0 0 0 0 2 1 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 3 J (a) (b) (c) (d) (e) Find the characteristic polynomial of T. Find the dot diagram corresponding to each eigenvalue of T. For which eigenvalues A;, if any, docs EA, = KA/? For each eigenvalue A*, find the smallest positive integer pi for which KAi = N((T - AJ)^). Compute the following numbers for each i, where Uj denotes the restriction of T - Ajl to KA, . (i) rank(Ui) (ii) rank(U2) (iii) nullity(U?) (iv) nullity(U2) 4. For each of the matrices A that follow, find a Jordan canonical form J and an invertible matrix Q such that .7 = Q~1AQ. Notice that the matrices in (a), (b), and (c) are those used in Example 5. (a) A = 1 2 - 1 2 - 1 2 1 4/ (c) A = (d) A ( 0 - 2 - 2 V - 2 Sec. 7.2 The Jordan Canonical Form II 511 5. For each linear operator T, find a Jordan canonical form J of T and a Jordan canonical basis 0 for T. (a) V is the real vector space of functions spanned by the set of real-valued functions {et,tet,t2et,e2t}, and T is the linear operator on V defined by T( / ) = / ' . (b) T is the linear operator on ?3(R) defined by T(/(a:)) = xf'(x). (c) T is the linear operator on P-$(R) defined by T(f(x)) = f'(x) + 2f(x). (d) T is the linear operator on M2x2(i?) defined by T ( A ) - ( ty-A-A*. (e) T is the linear operator on M2x2(i?) defined by T(A)=(l ty(A-A% (f) V is the vector space of polynomial functions in two real variables x and y of degree at most 2, as defined in Example 4, and T is the linear operator on V defined by T(f(x,y)) = f(x,y) + ^f(x,y). 6. Let A be an n x n matrix whose characteristic polynomial splits. Prove that A and A1 have the same Jordan canonical form, and conclude that A and A* are similar. Hint: For any eigenvalue A of A and A1 and any positive integer r, show that rank((A Al)r) = rank((Al Al)r). 7. Let A be an n x n matrix whose characteristic polynomial splits, 7 be a cycle of generalized eigenvectors corresponding to an eigenvalue A, and W be the subspace spanned by 7. Define 7 ' to be the ordered set obtained from 7 by reversing the order of the vectors in 7. (a) Prove that [Tw]7' = ([Tw]7) (b) Let J be the Jordan canonical form of A. Use (a) to prove that J and J* are similar. (c) Use (b) to prove that A and At are similar. 8. Let T be a linear operator on a finite-dimensional vector space, and suppose that the characteristic polynomial of T splits. Let 0 be a Jordan canonical basis for T. (a) Prove that for any nonzero scalar c, {ex: x 0} is a Jordan canon-ical basis for T. 512 Chap. 7 Canonical Forms (b) Suppose that 7 is one of the cycles of generalized eigenvectors that forms 0, and suppose that 7 corresponds to the eigenvalue A and has length greater than 1. Let x be the end vector of 7, and let y be a nonzero vector in EA- Let 7' be the ordered set obtained from 7 by replacing x by x + y. Prove that 7' is a cycle of generalized eigenvectors corresponding to A, and that if 7' replaces 7 in the union that defines 0, then the new union is also a Jordan canonical basis for T. (c) Apply (b) to obtain a Jordan canonical basis for LA, where A is the matrix given in Example 2, that is different from the basis given in the example. 9. Suppose that a dot diagram has k columns and m rows with pj dots in column j and r* dots in row i. Prove the following results. (a) rn p and k = r. (b) Pj = max {i: Ti > j} for 1 < j < k and r^ = max {j: pj > i} for 1 < i < rn. Hint: Use mathematical induction on m. (c) n > r2 > > rm. (d) Deduce that the number of dots in each column of a dot diagram is completely determined by the number of dots in the rows. 10. Let T be a linear operator whose characteristic polynomial splits, and let A be an eigenvalue of T. (a) Prove that dim(KA) is the sum of the lengths of all the blocks corresponding to A in the Jordan canonical form of T. (b) Deduce that EA = KA if and only if all the Jordan blocks corre-sponding to A are l x l matrices. The following definitions arc used in Exercises 11 19. Definitions. A linear operator T on a vector space V is called nilpotcnt if Tp = To for some positive integer p. An nxn matrix A is called nilpotent if Ap O for some positive integer p. 11. Let T be a linear operator on a finite-dimensional vector space V, and let 0 be an ordered basis for V. Prove that T is nilpotcnt if and only if T]j3 is nilpotent. 12. Prove that any square upper triangular matrix with each diagonal entry equal to zero is nilpotent. 13. Let T be a nilpotent operator on an n-dimensional vector space V, and suppose that p is the smallest positive integer for which Tp = T0. Prove the following results. (a) N(P) C N(T'+1) for every positive integer i. Sec. 7.2 The Jordan Canonical Form II 513 (b) There is a sequence of ordered bases 0, p,..., 0P such that 0i is a basis for N(T*) and 0i+ contains 0i for 1 < i < p 1. (c) Let 0 = 0P be the ordered basis for N(TP) = V in (b). Then [T]^ is an upper triangular matrix with each diagonal entry equal to zero. (d) The characteristic polynomial of T is (l)ntn. Hence the charac-teristic polynomial of T splits, and 0 is the only eigenvalue of T. 14. Prove the converse of Exercise 13(d): If T is a linear operator on an n-dimensional vector space V and (l)ntn is the characteristic polynomial of T, then T is nilpotent. 15. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. 16. Let T be a nilpotent linear operator on a finite-dimensional vector space V. Recall from Exercise 13 that A = 0 is the only eigenvalue of T, and hence V = KA- Let 0 be a Jordan canonical basis for T. Prove that for any positive integer i, if we delete from 0 the vectors corresponding to the last i dots in each column of a dot. diagram of 0. the resulting set is a basis for R(T*). (If a column of the dot diagram contains fewer than i dots, all the vectors associated with that column arc removed from 0.) 17. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let Ai, A 2 , . . . , Afc be the distinct eigenvalues of T. Let S: V * V be the mapping defined by S(x) = Ai?;j + X2v2 H h Afc-Ufc, where, for each i, Vi is the unique vector in KA, such that x v + V2 H -Vfc- (This unique representation is guaranteed by Theorem 7.3 (p. 486) and Exercise 8 of Section 7.1.) (a) Prove that S is a diagonalizable linear operator on V. (b) Let U = T S. Prove that U is nilpotent and commutes with S, that is, SU = US. 18. Let T be a linear operator on a finite-dimensional vector space V, and let J be the Jordan canonical form of T. Let D be the diagonal matrix whose diagonal entries are the diagonal entries of J, and let M = J D. Prove the following results. (a) M is nilpotent. (b) MD = DM. 514 Chap. 7 Canonical Forms (c) If p is the smallest positive integer for which Mp O, then, for any positive integer r < p, r = >r + rDr~1M + r{'r~V)Dr-2M2 + + rDMr~l + Mr, and, for any positive integer r > p, jr = Dr + rDr~lM + r^~^Dr-2M2 + 2! + r! (r-p-rl)(p-l) Dr~p+1Mp-1. 19. Let J = (X 1 0 0 0 A 1 0 0 0 A 0 0 0 0 1 0 0 0 A/ be the m x m Jordan block corresponding to A, and let N = J XIm. Prove the following results: (a) N m = O, and for 1 < r < m, 0 otherwise. (b) For any integer r >m, 1 _x r ( r - l ) r ( r - l ) - - - ( r - m + 2 ) x r _ m + 1 ^ 2! J r = Ar rAT 0 A? ( m - 1 ) ! rm+l rX r - l r ( r - l ) - - - ( r - m + 3) . r _ m + 2 ( m - 2 ) ! 0 0 0 AT (c) lim Jr exists if and only if one of the following holds: T>00 (i) |A| < 1. (ii) A = 1 and m = 1. - w . Sec. 7.2 The Jordan Canonical Form II 515 (Note that lim Ar exists under these conditions. See the discus-r*oo sion preceding Theorem 5.13 on page 285.) Furthermore, lim Jr roo is the zero matrix if condition (i) holds and is the l x l matrix (1) if condition (ii) holds, (d) Prove Theorem 5.13 on page 285. The following definition is used in Exercises 20 and 21. Definition. For any A G Mn X n(C), define the norm of A by A = max{|j4jj|: 1 < i,j < n}. 20. Let A, B G M n x n (C) . Prove the following results. (a) ||A|| > 0 and ||A|| = 0 if and only if A = O. (b) cA = |c|||i4|| for any scalar c. (c) P + S | |< ||A|| + ||||. (d) | |AB| |oo absolute value 1. (e) Theorem 5.20(a) using (c) and Theorem 5.19. The next exercise requires knowledge of absolutely convergent series as well as the definition of eA for a matrix A. (See page 312.) 22. Use Exercise 20(d) to prove that eA exists for every A G M n x n (C) . 23. Let x' Ax be a system of n linear differential equations, where x is an n-tuple of differentiable functions x(t),X2(t),... ,xn(t) of the real variable t, and A is an n x n coefficient matrix as in Exercise 15 of Section 5.2. In contrast to that exercise, however, do not assume that A is diagonalizable, but assume that the characteristic polynomial of A splits. Let Ai, A 2 , . . . , Afc be the distinct eigenvalues of A. 516 Chap. 7 Canonical Forms (a) Prove that if u is the end vector of a cycle of generalized eigenvec-tors of LA of length p and u corresponds to the eigenvalue A;, then for any polynomial f(t) of degree less than p, the function ex*[f(t)(A - Xilf-1 + f'(t)(A - Xi)p-2 + + / ^ ( i O J t i is a solution to the system x' = Ax. (b) Prove that the general solution to x' = Ax is a sum of the functions of the form given in (a), where the vectors u are the end vectors of the distinct cycles that constitute a fixed Jordan canonical basis for LA-24. Use Exercise 23 to find the general solution to each of the following sys-tems of linear equations, where x, y, and z are real-valued differentiable functions of the real variable t. x' = 2x + y x' = 2x + y (a) y'= 2y- z (b) y'= 2y + z z' = 3z z'= 2z 7.3 THE MINIMAL POLYNOMIAL The Cayley-Hamilton theorem (Theorem 5.23 p. 317) tells us that for any linear operator T on an n-dimensional vector space, there is a polynomial f(t) of degree n such that f(T) = To, namely, the characteristic polynomial of T. Hence there is a polynomial of least degree with this property, and this degree is at most n. If g(t) is such a polynomial, we can divide g(t) by its leading coefficient to obtain another polynomial p(t) of the same degree with leading coefficient 1, that is, p(t) is a monic polynomial. (See Appendix E.) Definition. Let T be a linear operator on a finite-dimensional vector space. A polynomial p(t) is called a minimal polynomial ofT if p(t) is a monic polynomial of least positive degree for which p(T) To. The preceding discussion shows that every linear operator on a finite-dimensional vector space has a minimal polynomial. The next result shows that it is unique. Theorem 7.12. Let p(t) be a minimal polynomial of a linear operator T on a finite-dimensional vector space V. (a) For any polynomial g(t), if g(T) = To, thenp(t) divides g(t). In partic-ular, p(t) divides the characteristic polynomial ofT. (b) The minimal polynomial of T is unique. Proof, (a) Let g(t) be a polynomial for which g(T) = TQ. By the division algorithm for polynomials (Theorem E.l of Appendix E, p. 562), there exist polynomials q(t) and r(t) such that 9(t) = q(t)p(t)+r(t), (1) Sec. 7.3 The Minimal Polynomial 517 where r(t) has degree less than the degree of p(i). Substituting T into (1) and using that g(T) p(T) T0, we have r(T) = To. Since r(t) has degree less than p(t) and p(t) is the minimal polynomial of T, r(t) must be the zero polynomial. Thus (1) simplifies to g(t) q(t)p(t), proving (a). (b) Suppose that p(t) and p2() are each minimal polynomials ofT. Then P(t) divides p2() by (a). Since p(t) and P2(t) have the same degree, we have that p2(t) = cpi(t) for some nonzero scalar c. Because Pi(t) and p2(i) are monic, c = 1; hence P(t) = p2(i). D The minimal polynomial of a linear operator has an obvious analog for a matrix. Definition. Let A Mn X n(F). The minimal polynomial p(t) of A is the monic polynomial of least positive degree for which p(A) = O. The following results are now immediate. Theorem 7.13. Let T be a linear operator on a finite-dimensional vector space V, and let 0 be an ordered basis for V. Then the minimal polynomial ofT is the same as the minimal polynomial of [T]p. Proof. Exercise. [~] Corollary. For any A G Mnxn(F), the minimal polynomial of A is the same as the minimal polynomial of LA Proof. Exercise. I | In view of the preceding theorem and corollary, Theorem 7.12 and all subsequent theorems in this section that are stated for operators are also valid for matrices. For the remainder of this section, we study primarily minimal polynomials of operators (and hence matrices) whose characteristic polynomials split. A more general treatment of minimal polynomials is given in Section 7.4. Theorem 7.14. Let T be a linear operator on a finite-dimensional vector space V, and let p(t) be the minimal polynomial of T. A scalar X is an eigenvalue ofT if and only ifp(X) = 0. Hence the characteristic polynomial and the minimal polynomial of T have the same zeros. Proof. Let f(t) be the characteristic polynomial of T. Since p(t) divides f(t), there exists a polynomial q(t) such that f(t) = q(t)p(t). If A is a zero of p(t), then f(X) = q(X)p(X) = q(X).0 = 0. So A is a zero of f(t); that is, A is an eigenvalue of T. 518 Chap. 7 Canonical Forms Conversely, suppose; that A is an eigenvalue of T, and let x G V be an eigenvector corresponding to A. By Exercise 22 of Section 5.1, we have 0 = To(aO=p(T)(aO=p(A)a;. Since x ^ 0. it follows that p(X) = 0, and so A is a zero of p(t). I The following corollary is immediate. Corollary. Let T be a linear operator on a finite-dimensional vector space V with minimal polynomial p(t) and characteristic polynomial f(t). Suppose that f(t) factors as f(t) = (1-t)ni(2-t)n>-..(k-t)n'>, where X, A 2 , . . . , Afc are the distinct eigenvalues ofT. Then there exist inte-gers mi, m 2 , . . . ,nifc such that 1 < m < rii for all i and p ( t ) ( t - A 1 ) m ( t - A a ) m ( - A*)*. Example 1 We compute the minimal polynomial of the matrix 4 = Since A has the characteristic polynomial (3-t - 1 0 / ( t ) = det 0 2-t 0 = -(t-2)2{t~3), i - 1 2 - 7 the minimal polynomial of A must be either (t 2)(t 3) or (t 2)2(t 3) by the corollary to Theorem 7.14. Substituting A into p(t) = (t - 2)(t - 3), we find that p(A) O; hence p(t) is the minimal polynomial of A. Example 2 Let T be the linear operator on R2 defined by T(a, b) = (2a + 56,6a + b) and 0 be the standard ordered basis for R2. Then '2 ;> [T]p = 6 1 and hence the characteristic polynomial of T is / ( * ) = d e t ( 2 ~ ' l l t ) = ( * - 7 ) ( * + 4). Thus the minimal polynomial of T is also (t 7)(t + 4). Sec. 7.3 The Minimal Polynomial 519 Example 3 Let D be the linear operator on P2(/?) defined by D(g(x)) g'(x), the deriva-tive of g(x). We compute the minimal polynomial of T. Let 0 be the standard ordered basis for P2(i?). Then /(> 1 0 [O]0 = 0 0 2 , o 0 0 / and it follows that the characteristic polynomial of D is t3. So by the corollary to Theorem 7.14, the minimal polynomial of D is t, t2, or t3. Since D2(a;2) = 2 ^ 0, it follows that D2 ^ T0; hence the minimal polynomial of D must be t3. In Example 3, it is easily verified that P2(i?) is a D-cyclic subspace (of itself). Here the minimal and characteristic polynomials are of the same degree. This is no coincidence. Theorem 7.15. Let T be a linear operator on an n-dimcnsional vector space V such that V is a T -cyclic subspace of itself. Then the characteristic polynomial f(t) and the minimal polynomial p(t) have the same degree, and hence f(t) = (-l)np(t). Proof. Since V is a T-cyclic space, there exists an ,7; G V such that 0 = {x,T(x),...,Tn-1(x)} is a basis for V (Theorem 5.22 p. 315). Let g(t) = a0-ra1t-r--- + aktk, be a polynomial of degree k < n. Then afc 7^ 0 and g(T)(x) = a0x + axT(x) + + o,kTk(x), and so g(T)(x) is a linear combination of the vectors of 0 having at least one nonzero coefficient, namely, afc. Since 0 is linearly independent, it follows that g(T)(x) ^ 0; hence g(T) / To. Therefore the minimal polynomial of T has degree n, which is also the degree of the characteristic polynomial of T. Theorem 7.15 gives a condition under which the degree of the minimal polynomial of an operator is as large as possible. We now investigate the other extreme. By Theorem 7.14, the degree of the minimal polynomial of an operator must be greater than or equal to the number of distinct eigenvalues of the operator. The next result shows that the operators for which the degree of the minimal polynomial is as small as possible are precisely the diagonalizable operators. 520 Chap. 7 Canonical Forms Theorem 7.16. Let T be a linear operator on a finite-dimensional vector space V. Then T is diagonalizable if and only if the minimal polynomial ofT is of the form p(t) = (t-X1)(t-X2)---(t-Xk), where Ai. A2 AA- are the distinct eigenvalues ofT. Proof. Suppose that T is diagonalizable. Let Ai, A2 , Afc be the distinct eigenvalues of T, and define p(t) = (t-1)(t-2)(t- Afe). By Theorem 7.14, p(t) divides the minimal polynomial of T. Let 0 {vi,V2,- ,Vn} be a basis for V consisting of eigenvectors of T. and con-sider any '.',: G 0. Then (T Xj)(Vi) = 0 for some eigenvalue Ay. Since t - A? divides p(i), there is a polynomial qj(t) such that p(t) = qj(t)(t - Xj). Hence p(T)(vi) = qj(T)(T-Xj)(vi) = 0. It follows that p(T) = To, since p(T) takes each vector in a basis for V into 0. Therefore p(t) is the minimal polynomial of T. Conversely, suppose that there are distinct scalars Ai, A 2 , . . . , Afc such that the minimal polynomial p(t) of T factors as p(t) = (t-X1)(t-X2)(*- Afc). By Theorem 7.14. the A/'s are eigenvalues of T. We apply mathematical induction on n = diin(V). Clearly T is diagonalizable for n = 1. Now assume that T is diagonalizable whenever dim(V) < n for some n > 1, and lei dim(V) = n and W = R(T - A/,-1). Obviously W ^ V, because Afc is an eigenvalue of T. If W = {0}. then T = Afc I. which is clearly diagonalizable. So suppose that 0 < diin(W) < n. Then W is T-invariant, and for any x G W, (T - A,I)(T - A2I) (T - Afc_,!)(:/;) = 0. It follows that the minimal polynomial of Tw divides the polynomial (/ Ai)(r - A2) (/ Afc_i). Hence by the induction hypothesis, Tw is diagonalizable. Furthermore, Afc is not an eigenvalue of Tw by Theorem 7.14. Therefore W n N(T - AAI) = {()}. Now let ft = {ci.r,.... , vm) be a ba-sis for W consisting of eigenvectors of Tw (and hence of T), and let 02 = {w, W2, ,wp} be a basis for N(T - Afcl), the eigenspace of T corresponding to Afc. Then 0 and 02 are disjoint by the previous comment. Moreover, m+p = n by the dimension theorem applied to T - Afcl. We show that Q = 0i U 02 is linearly independent. Consider scalars a ] , a 2 , . . . , a m and b.b2.... ,bv such that aV + a2t'2 ,t'm + b u- + b2tiSec. 7.3 The Minimal Polynomial Let 521 x = 22 aiVi and */ = ! >< (D, i=l Then x G W, y G N(T - Afcl), and x + y = 0. It follows that x = -y 522 Chap. 7 Canonical Forms EXERCISES 1. Label the following statements as true or false. Assume that all vector spaces are finite-dimensional. (a) Every linear operator T has a polynomial p(t) of largest degree for which 7->(T) = TQ. (b) Every linear operator has a unique minimal polynomial. (c) The characteristic polynomial of a linear operator divides the min-imal polynomial of that operator. (d) The minimal and the characteristic polynomials of any diagonal-izable operator are equal. (e) Let T be a linear operator on an n-dimensional vector space V, p(t) be the minimal polynomial of T, and f(t) be the characteristic polynomial of T. Suppose that f(t) splits. Then f(t) divides p(t)n-(f) The minimal polynomial of a linear operator always has the same degree as the characteristic polynomial of the operator. (g) A linear operator is diagonalizable if its minimal polynomial splits. (h) Let T be a linear operator on a vector space V such that V is a T-cyclic subspace of itself. Then the degree of the minimal poly-nomial of T equals dim(V). (i) Let T be a linear operator on a vector space V such that T has n distinct eigenvalues, where n dim(V). Then the degree of the minimal polynomial of T equals n. 2. Find the minimal polynomial of each of the following matrices. (a) (c) 2 1 4 1 1 1 2 j -14 - 4 - 6 5 2 4 (b) 1 1 0 1 3. 4. 5. For each linear operator T on V, find the minimal polynomial of T. (a) V = R2 and T(o, b) = (a + b, a - b) (b) V = P2(R) and T(g(x)) = g'(x) + 2g(x) (c) V = P2(R) and T(f(x)) = -xf'(x) + f'(x) + 2f(x) (d) V = MnXn(i?) and T(A) = A*. Hint: Note that T2 = I. Determine which of the matrices and operators in Exercises 2 and 3 are diagonalizable. Describe all linear operators T on R2 such that T is diagonalizable and T3 _ 2T2 + T = T0. Sec. 7.3 The Minimal Polynomial 6. Prove Theorem 7.13 and its corollary. 7. Prove the corollary to Theorem 7.14. 523 8. Let T be a linear operator on a finite-dimensional vector space, and let p(t) be the minimal polynomial of T. Prove the following results. (a) T is invertible if and only if p(0) - 0. (b) If T is invertible and p(t) = tn + a n _i n _ 1 + + ait + a0 , then T - 1 = - (T'- 1 + a n _ , T n - 2 + + a 2 T + a , I ) . a0 9. Let T be a diagonalizable linear operator on a finite-dimensional vector space V. Prove that V is a T-cyclic subspace if and only if each of the eigenspaces of T is one-dimensional. 10. Let T be a linear operator on a finite-dimensional vector space V, and suppose that W is a T-invariant subspace of V. Prove that the minimal polynomial of Tw divides the minimal polynomial of T. 11. Let g(t) be the auxiliary polynomial associated with a homogeneous lin-ear differential equation with constant coefficients (as defined in Section 2.7), and let V denote the solution space of this differential equation. Prove the following results. (a) V is a D-invariant subspace, where D is the differentiation operator on C. (b) The minimal polynomial of Dv (the restriction of D to V) is g(t). (c) If the degree of g(t) is n, then the characteristic polynomial of Dv is(-l)ng(t). Hint: Use Theorem 2.32 (p. 135) for (b) and (c). 12. Let D be the differentiation operator on P(/?), the space of polynomials over R. Prove that there exists no polynomial g(t) for which ; 524 Chap. 7 Canonical Forms 14. Let T be linear operator on a finite-dimensional vector space V, and let Wi and W2 be T-invariant subspaces of V such that V = W| W2. Suppose that pi(t) and p2(t) are the minimal polynomials of Tw, and Tw2, respectively. Prove or disprove; that p{t)p2(t) is the minimal polynomial of T. Exercise 15 uses the following definition. Definition. Let T be a linear operator on a linite-dimensional vector space V, and let x be a nonzero vector in V. The polynomial p(t) is called a T-annihilator of x if p(t) is a monic polynomial of least degree for which p(T)(x) = 0. 15. * Let T be a linear operator on a finite-dimensional vector space V, and let x be a nonzero vector in V. Prove the following results. (a) The vector x has a unique T-annihilator. (b) The T-annihilator of X divides any polynomial git) for which 9(T) = T(). (c) If p(t) is the T-annihilator of x and W is the T-cyclic subspace generated by x, then />(Sec. 7.4 The Rational Canonical Form 525 uniquely as f(t) = (-i)n(Mt)r(526 Chap. 7 Canonical Forms 0X. Let A be the matrix representation of the restriction of T to Cx relative to the ordered basis 0X. Recall from the proof of Theorem 5.22 that A / o 1 0 0 0 1 0 0 0 -ao - a i - a 2 xo 0 where aox + aT(x) + 1 -a fc- i / ak-lTk-l{x) + Tk(x) = 0. Furthermore, the characteristic polynomial of A is given by det(i4 - ti) = (- l) f c(a0 + axt + + ak-it k- The matrix A is called the companion matrix of the monic polynomial h(t) = ao + ai/. + + ak-ifk~1 + tk. Every monic polynomial has a com-panion matrix, and the characteristic polynomial of the companion matrix of a monic polynomial g(t) of degree k is equal to ( l)kg(t). (See Exercise 19 of Section 5.4.) By Theorem 7.15 (p. 519), the monic polynomial h(t) is also the minimal polynomial of A. Since A is the matrix representation of the restriction of T to Cx, h(t) is also the minimal polynomial of this restriction. By Exercise 15 of Section 7.3, h(t) is also the T-annihilator of x. It is the object of this section to prove that for every linear operator T on a finite-dimensional vector space V, there exists an ordered basis 0 for V such that the matrix representation [T]^ is of the form (Cl O o c2 o o o Cr) where each C7; is the companion matrix of a polynomial ((t))m such that ~ = Sec. 7.4 The Rational Canonical Form 527 Proof. By Exercise 15(b) of Section 7.3, ((t))p divides the minimal poly-nomial of T. Therefore 4>(t) divides the minimal polynomial of T. Further-more, x G K,p by the definition of K ,^. | Theorem 7.17. Let T be a linear operator on a finite-dimensional vector space V, and let 0 be an ordered basis for V. Then 0 is a rational canonical basis for T if and only if 0 is the disjoint union of T-cyclic bases 0Vi, where each Vi lies in K^ for some irreducible monic divisor 528 Chap. 7 Canonical Forms The rational canonical form C of the operator T in Example 1 is con-structed from matrices of the form C, each of which is the companion matrix of some power of a monic irreducible divisor of the characteristic polynomial of T. Furthermore, each such divisor is used in this way at least once. In the course of showing that, every linear operator T on a finite dimen-sional vector space has a rational canonical form C, we show that the com-panion matrices C, that constitute C are always constructed from powers of the monic irreducible divisors of the characteristic polynomial of T. A key role in our analysis is played by the subspaces K^, where 4>(t) is an irreducible monic divisor of the minimal polynomial of T. Since the minimal polynomial of an operator divides tin1 characteristic polynomial of the operator, every ir-reducible divisor of the former is also an irreducible divisor of the latter. We eventually show that the converse is also true: that is. the minimal polynomial and the characteristic polynomial have the same irreducible divisors. We begin with a result that lists several properties of irreducible divisors of the minimal polynomial. The reader is advised to review the definition of T-annihilator and the accompanying Exercise 15 of Section 7.3. Theorem 7.18. Let T lie a linear operator on a finite-dimensional vector space V. and suppose that p(t) = (Mt))mi(Mt)r3---(Mt))mk is the minimal polynomial ofT. where the i(t)*s (1 < i < k) arc the distinct irreducible monic factors of pit) and the nij 's are positive integers. Then the following statements are true. (a) K0( i.s a nonzero T-invariant subspace of V for each i. (b) If x is a nonzero vector in some K^,. then the T-annihilator of x is of the form ((j>i(t))p for some integer p. (c) K0, nKlh = {()} fori ^j. . (d) K0, is invariant under (f>j(T) for i ^ j, and the restriction of 0;(T) to K(j>j is one-to-one and onto. (e) K,,' - N((0,(T))' 1. (a) The proof that Kfl>i is a T-invariant subspace of V is left as an exer-cise. Let fi(t) be the polynomial obtained from p(t) by omitting the factor (0;(/))''. To prove that K^. is nonzero, first observe that / ,( /) is a proper di-visor of pit): therefore there exists a vector z G V such that x fi(T)(z) ^ 0. Then x G K,.,( because (i(T))mi(x) = (d>i(T))mifi(T)(z)=p(T)(z) = 0. (b) Assume the hypothesis. Then (i(T))q(x) = 0 for some positive in-teger q. Hence the T-annihilator of x divides (4>%(t))q by Exercise 15(b) of Section 7.3. and the result follows. Sec. 7.4 The Rational Canonical Form 529 (c) Assume i ^ j. Let x G Ki(t) and i(t) and j(t) are relatively prime (see Appendix E). We conclude that x = 0. (d) Assume / ^ j. Since K530 Chap. 7 Canonical Forms is the minimal polynomial of T, where the 0j 's (1 < i < fc) are the dis-tinct irreducible monic factors ofp(t) and the 7m's are positive integers. For 1 < i < k, let Si be a linearly independent subset of K0i . Then (a) 5j n 5j = 0 for i ^ j (b) S U S2 U U Sfc is linearly independent. Proof. If A: 1, then (a) is vacuously true and (b) is obvious. Now suppose that k > 1. Then (a) follows immediately from Theorem 7.18(c). Furthermore, the proof of (b) is identical to the proof of Theorem 5.8 (p. 267) with the eigenspaces replaced by the subspaces K^,. i In view of Theorem 7.19. we can focus on bases of individual spaces of the form K(/)(/), where 0(f) is an irreducible monic divisor of the minimal polynomial of T. The next several results give us ways to construct bases for these spaces that are unions of T-cyclic bases. These results serve the dual purposes of leading to the existence theorem for the rational canonical form and of providing methods for constructing rational canonical bases. For Theorems 7.20 and 7.21 and the latter's corollary, we fix a linear operator T on a finite-dimensional vector space V and an irreducible monic divisor 6(t) of the minimal polynomial of T. Theo rem 7.20. Let v, v2,..., vk be distinct vectors in K,p such that Si =0VlU 0V2 U U 0Vk is linearly independent. For each i, choose wi G V such that 0(T)(w^) = t'j. Then S2 = 0un U 0W2 U U 0Wk is also linearly independent. Proof. Consider any linear combination of vectors in 52 that sums to zero, say, < = 1 j=0 For each i. let /,-(/) be the polynomial defined by (3) / i ( t ) = y w . 1=0 Then (3) can be rewritten as / < ( T ) ( t m ) = 0. (4) i= Sec. 7.4 The Rational Canonical Form Apply 0(T) to both sides of (4) to obtain 531 I > 0 O / < ( T ) ( t i ; 0 = J > 0 > ( T ) ( u , . ) = X > ( T ) ( 0 = i=i i=l i= This last sum can be rewritten as a linear combination of the vectors in S so that each /(T)(j) is a linear combination of the vectors in 0Vi. Since Si is linearly independent, it follows that fi(T)(vi) = 0 for all*. Therefore the T-annihilator of vi divides fi(t) for all i. (See Exercise 15 of Section 7.3.) By Theorem 7.18(b), 0(f) divides the T-annihilator of Vi, and hence 0(f) divides fi(t) for all i. Thus, for each i, there exists a polynomial gi(t) such that fi(t) = 532 Chap. 7 Canonical Forms where d is the degree of 0(f). Then z G Cx D W, and hence C2 C Cx n W. Suppose that z ^ 0. Then 2 has 0(f) as its T-annihilator, and therefore d = dim(Q) < dim(Cx n W ) < dim(Cx) = d. It follows that Cx PI W = Cx, and consequently x G W, contrary to hypothesis. Therefore z = 0, from which it follows that bj 0 for all j. Since 0 is linearly independent, it follows that a^ = 0 for all i. Thus 0 U 0X is linearly independent. (b) Suppose that W does not contain N(0(T)). Choose a vector tui G N(0(f)) that is not in W. By (a), 0 = 0 U 0Wl is linearly independent. Let Wi = span(/?i). If Wi does not contain N(0(f)), choose a vector u;2 in N(0(f)), but not in Wi, so that /32 = 0 D0W2 = 0U0Wl Li0W2 is linearly inde-pendent. Continuing this process, we eventually obtain vectors w, w 2 , . . . , ws in N(0(T)) such that the union 0'=0U0WlU0W2U---U0Wa is a linearly independent set whose span contains N(0(T)). | Theorem 7.21. If the minimal polynomial ofT is of the form p(t) = (0(f) )m , then there exists a rational canonical basis for T. Proof. The proof is by mathematical induction on m. Suppose that m = 1. Apply (b) of the lemma to W = {0} to obtain a linearly independent subset of V of the form 0Vl U 0V2 U U 0Vk, whose span contains N(0(T)). Since V = N(0(T)), this set is a rational canonical basis for V. Now suppose that, for some integer m > 1, the result is valid whenever the minimal polynomial of T is of the form (0(T))fc, where k < m, and assume that the minimal polynomial of T is p(t) = (0(f) )m . Let r = rank(0(T)). Then R(0(T)) is a T-invariant subspace of V, and the restriction of T to this subspace has (0(f))m _ 1 as its minimal polynomial. Therefore we may apply the induction hypothesis to obtain a rational canonical basis for the restriction of T to R(T). Suppose that v, v 2 , . . . ,vk are the generating vectors of the T-cyclic bases that constitute this rational canonical basis. For each i, choose Wi in V such that Vi = cf>(T)(wi). By Theorem 7.20, the union 0 of the sets 0Wi is linearly independent. Let W = span(/3). Then W contains R(0(T)). Apply (b) of the lemma and adjoin additional T-cyclic bases Ai,fc+1,/?u,fc+2,... ,0W to 0, if necessary, where w^ is in N(0(T)) for i > k, to obtain a linearly independent set 0' = 0Wl U0W2 U U0Wk U U0Wa whose span W contains both W and N(0(T)). Sec. 7.4 The Rational Canonical Form 533 We show that W = V. Let U denote the restriction of 0(T) to W , which is 0(T)-invariant. By the way in which W was obtained from R(0(T)), it follows that R(U) = R(0(T)) and N(U) = N(0(T)). Therefore dim(W') = rank(U) + nullity(U) = rank(0(T)) + nullity(0(T)) = dim(V). Thus W = V, and 0' is a rational canonical basis for T. | Corollary. K0 has a basis consisting of the union of T-cyclic bases. Proof. Apply Theorem 7.21 to the restriction of T to K^. I We are now ready to study the general case. Theorem 7.22. Every linear operator on a finite-dimensional vector space has a rational canonical basis and, hence, a rational canonical form. Proof. Let T be a linear operator on a finite-dimensional vector space V, and let p(t) = (0i(f))mi(02(*))m2 (k(t))mk be the minimal polynomial of T, where the 0i(f)'s are the distinct irreducible monic factors of p(t) and mi > 0 for all i. The proof is by mathematical induction on k. The case k 1 is proved in Theorem 7.21. Suppose that the result is valid whenever the minimal polynomial contains fewer than k distinct irreducible factors for some k > 1, and suppose that p(t) contains k distinct factors. Let U be the restriction of T to the T-invariant subspace W = R((0fc(T)mfc), and let q(t) be the minimal polynomial of U. Then q(t) divides p(t) by Exercise 10 of Section 7.3. Furthermore, 0fc(f) does not divide q(t). For otherwise, there would exist a nonzero vector x G W such that 0fc(U)(:r) = 0 and a vector y G V such that x = (4>k{T))mk(y). It follows that (MT))mk + 1(y) = 0, and hence y G K ^ and x = f>fc(T))m*(y) = 0 by Theorem 7.18(e), a contradiction. Thus q(t) contains fewer than k distinct irreducible divisors. So by the induction hypothesis, U has a rational canonical basis 0 consisting of a union of U-cyclic bases (and hence T-cyclic bases) of vectors from some of the subspaces K0i, 1 < i < k 1. By the corollary to Theorem 7.21, K0fc has a basis 02 consisting of a union of T-cyclic bases. By Theorem 7.19, 0 and 02 are disjoint, and 0 = 0i U/?2 is linearly independent. Let s denote the number of vectors in 0. Then s = dim(R((0fc(T))Wfc)) + dim(K0fc) = rank((0 f c(T)r*) + nullity((0fc(T))TOfc) = n. We conclude that 0 is a basis for V. Therefore 0 is a rational canonical basis, and T has a rational canonical form. I 534 Chap. 7 Canonical Forms In our study of the rational canonical form, we relied on the minimal polynomial. We are now able to relate the rational canonical form to the characteristic polynomial. Theorem 7.23. Let T be a linear operator on an n-dimensional vector space V with characteristic polynomial /(f) = ( - i ) ( f c ( t ) r (Mt)r (Mt))nk, where the (pi(t)'s (I < i < k) are distinct irreducible monic polynomials and the Hi's are positive integers. Then the following statements are true. (a) 0i(f), 02(f) , . . . , 0fc(f) are the irreducible monic factors of the minimal polynomial. (b) For each i, dim(K0i) = diUi, where di is the degree of 0i(f). (c) If 0 is a rational canonical basis for T, then 0i = 0C K0i is a basis for K0i for each i. (d) If 7J is a basis for K0i for each i, then 7 = 71 U 72 U U 7fc is a basis for V. In particular, if each 7$ is a disjoint union of T-cyclic bases, then 7 is a rational canonical basis for T. Proof, (a) By Theorem 7.22, T has a rational canonical form C. By Exercise 40 of Section 5.4, the characteristic polynomial of C, and hence of T, is the product of the characteristic polynomials of the companion matrices that compose C. Therefore each irreducible monic divisor 0i(f) of /(f) divides the characteristic polynomial of at least one of the companion matrices, and hence for some integer p, (0j(f))p is the T-annihilator of a nonzero vector of V. We conclude that (0j(f))p, and so 0j(f), divides the minimal polynomial of T. Conversely, if 0(f) is an irreducible monic polynomial that divides the minimal polynomial of T, then 0(f) divides the characteristic polynomial of T because the minimal polynomial divides the characteristic polynomial. (b), (c), and (d) Let C T]p, which is a rational canonical form of T. Consider any i, (1 < i < k). Since /(f) is the product of the characteristic polynomials of the companion matrices that compose C, we may multiply those characteristic polynomials that arise from the T-cyclic bases in 0i to obtain the factor (0j(f))ni of /(f). Since this polynomial has degree nidi, and the union of these bases is a linearly independent subset 0i of K0i, we have nidi < dim(K0i). Furthermore, n = 2_.diUi, because this sum is equal to the degree of /(f). i=i Now let s denote the number of vectors in 7. By Theorem 7.19, 7 is linearly independent, and therefore = y^dini < / ]dim(K^i) = s < n. i=i i=i Sec. 7.4 The Rational Canonical Form 535 Hence n = s, and dini = dim(K()f,i) for all i. It follows that 7 is a basis for V and 0i is a basis for K^ for each i. I Uniqueness of the Rational Canonical Form Having shown that a rational canonical form exists, we are now in a po-sition to ask about the extent to which it is unique. Certainly, the rational canonical form of a linear operator T can be modified by permuting the T-cyclic bases that constitute the corresponding rational canonical basis. This has the effect of permuting the companion matrices that make up the rational canonical form. As in the case of the Jordan canonical form, we show that except for these permutations, the rational canonical form is unique, although the rational canonical bases are not. To simplify this task, we adopt the convention of ordering every rational canonical basis so that all the T-cyclic bases associated with the same irre-ducible monic divisor of the characteristic polynomial are grouped together. Furthermore, within each such grouping, we arrange the T-cyclic bases in decreasing order of size. Our task is to show that, subject to this order, the rational canonical form of a linear operator is unique up to the arrangement of the irreducible monic divisors. As in the case of the Jordan canonical form, we introduce arrays of dots from which we can reconstruct the rational canonical form. For the Jordan canonical form, we devised a dot diagram for each eigenvalue of the given operator. In the case of the rational canonical form, we define a dot diagram for each irreducible monic divisor of the characteristic polynomial of the given operator. A proof that the resulting dot diagrams are completely determined by the operator is also a proof that the rational canonical form is unique. In what follows, T is a linear operator on a finite-dimensional vector space with rational canonical basis 0; 0(f) is an irreducible monic divisor of the char-acteristic polynomial of T; 0Vl, 0V2,..., 0Vk are the T-cyclic bases of 0 that are contained in K^; and d is the degree of 0(f). For each j, let (0(f))Pj be the annihilator of Vj. This polynomial has degree dpj therefore, by Exercise 15 of Section 7.3, 0Vj contains dpj vectors. Furthermore, pi > p2 > > pk since the T-cyclic bases are arranged in decreasing order of size. We define the dot diagram of 0(f) to be the array consisting of k columns of dots with Pj dots in the jib. column, arranged so that the jfth column begins at the top and terminates after pj dots. For example, if k = 3, pi = 4, p2 = 2, and P3 = 2, then the dot diagram is Although each column of a dot diagram corresponds to a T-cyclic basis 536 Chap. 7 Canonical Forms 0Vi in K(/,, there are fewer dots in the column than there are vectors in the basis. Example 2 Recall the linear operator T of Example 1 with the rational canonical basis ii and the rational canonical form C [T]/j. Since there are two irreducible monic divisors of the characteristic polynomial of T, 0i(f) = t t + 3 and 02(f) = t + 1, there are two dot diagrams to consider. Because 0i(f) is the T-annihilator of V and 0Vl is a basis for K0 , . the dot diagram for 0i(f) consists of a single dot. The other two T cyclic bases, 0V3 and 0Vl, lie in K02. Since V3 has T-annihilator (02(f))2 and V7 has T-annihilator 02(f), in the dot diagram of 02(f) we have p = 2 and p2 = 1. These diagrams are as follows: Dot diagram for 0i(f) Dot diagram for -sit) = t2 + t + I. Suppose, furthermore, that the dot diagrams associated with these divisors are as follows: Diagram for 0i(f) Diagram for 2(t) Diagram for 03(f) Since the dot diagram for Oi (f) has two columns, it contributes two companion matrices to the rational canonical form. The first column has two dots, and therefore corresponds to the 2 x 2 companion matrix of (0i(f))2 = (f l)2. The second column, with only one dot, corresponds to the l x l companion matrix of 0i() = t 1. These two companion matrices are given by Ci = 0 - 1 1 2 and C2 = (). The dot diagram for 02(f) = f2 + 2 consists of two columns, each containing a single dot; hence this diagram contributes two copies of the 2 x 2 companion z MET Sec. 7.4 The Rational Canonical Form matrix for 02(f), namely, C-x ~ CA 537 The dot diagram for 03(f) = f2 4- f + 1 consists of a single column with a single dot contributing the single 2 x 2 companion matrix C, = 0 - 1 1 - l y Therefore the rational canonical form of T is the 9 x 9 matrix C = (Cx 0 O O o O C2 O O O O 0 c3 o o o o o c4 o o o O O C5J ( o 1 0 0 0 0 0 0 v 0 - 1 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 - 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 - 2 0 0 0 0 0 0 0 0 0 0 0 1 o > 0 0 0 0 0 0 - 1 -1 J We return to the general problem of finding dot diagrams. As we did before, we fix a linear operator T on a finite-dimensional vector space and an irreducible monic divisor 0(f) of the characteristic polynomial of T. Let U denote the restriction of the linear operator 0(T) to K^. By Theorem 7.18(d), Jq = TQ for some positive integer q. Consequently, by Exercise 12 of Sec-tion 7.2, the characteristic polynomial of U is ( - l ) w f m , where m = dim(K^). Therefore K^ is the generalized eigenspace of U corresponding to A = 0, and U has a Jordan canonical form. The dot diagram associated with the Jordan canonical form of U gives us a key to understanding the dot diagram of T that is associated with 0(f). We now relate the two diagrams. Let 0 be a rational canonical basis for T, and 0Vl, 0V2,..., 0Vk be the T-cyclic bases of 0 that are contained in K0. Consider one of these T-cyclic bases 0Vj, and suppose again that the T-annihilator of Vj is (4>(t))Pj. Then 0Vj consists of dpj vectors in 0. For 0 < i < d, let 7; be the cycle of generalized eigenvectors of U corresponding to A = 0 with end vector TZ(VJ), 538 Chap. 7 Canonical Forms where T(Vj) = bj. Then H = {((T)y,J-lVivj). (0 (T) )^ - 2 T' ( r , ) , . . , (Sec. 7.4 The Rational Canonical Form 539 replaced by the union otj of d cycles of generalized eigenvectors of U, each of length Pj, which becomes part of the Jordan canonical basis for U. In effect, otj determines d columns each containing pj dots in D2- So each column in Di determines d columns in D2 of the same length, and all columns in JD2 are obtained in this way. Alternatively, each row in D2 has d times as many dots as the corresponding row in D. Since Theorem 7.10 (p. 500) gives us the number of dots in any row of D2, we may divide the appropriate expression in this theorem by d to obtain the number of dots in the corresponding row of D. Thus we have the following result. Theorem 7.24. Let T be a linear operator on a finite-dimensional vector space V, let 0(f) be an irreducible monic divisor of the characteristic poly-nomial ofT of degree d, and let r ; denote the number of dots in the ith row of the dot diagram for 0(f) with respect to a rational canonical basis for T. Then (a) n = i[dim(V) - rank(0(T))] (b) n = i[rank((0(T)) i-1) - rank((0(T))i)l for % > 1. d Thus the dot diagrams associated with a rational canonical form of an op-erator are completely determined by the operator. Since the rational canoni-cal form is completely determined by its dot diagrams, we have the following uniqueness condition. Corollary. Under the conventions described earlier, the rational canonical form of a linear operator is unique up to the arrangement of the irreducible monic divisors of the characteristic polynomial. Since the rational canonical form of a linear operator is unique, the poly-nomials corresponding to the companion matrices that determine this form are also unique. These polynomials, which are powers of the irreducible monic divisors, are called the elementary divisors of the linear operator. Since a companion matrix may occur more than once in a rational canonical form, the same is true for the elementary divisors. We call the number of such occurrences the multiplicity of the elementary divisor. Conversely, the elementary divisors and their multiplicities determine the companion matrices and, therefore, the rational canonical form of a linear operator. Example 4 Let 0 = {ex cos 2x, ex sin 2x, xex cos 2x, xex sin 2x} 540 Chap. 7 Canonical Forms ( 1 - 2 0 ^ 0 2 1 0 0 1 0 0 1 1 2 - 2 V be viewed as a subset of J-(R, R), the space of all real-valued functions defined on R, and let V = span (/if). Then V is a four-dimensional subspace oiT(R, R), and 0 is an ordered basis for V. Let D be the linear operator on V defined by D(y) = y', the derivative of y, and let A [D]^. Then A = and the characteristic polynomial of D, and hence of A, is /(f) = ( f 2 - 2 f + 5)2. Thus 0(f) = f2 2f -f-5 is the only irreducible monic divisor of /(f). Since 0(/,) has degree 2 and V is four-dimensional, the dot diagram for 0(f) contains only two dots. Therefore the dot diagram is determined by ri , the number of dots in the first row. Because ranks are preserved under matrix representations, we can use A in place of D in the formula given in Theorem 7.24. Now 4>(A) = (0 0 0 4 0 0 - 4 0 0 0 0 0 0 0 0 o / and so r1 = l [4-rank(0(A))] = i [ 4 - 2 ] = l. It follows that the second dot lies in the second row, and the dot diagram is as follows: Hence V is a D-cyclic space generated by a single function with D-annihilator (0(f))2. Furthermore, its rational canonical form is given by the companion matrix of (0(f))2 = f4 - 4i3 + 14f2 - 20f + 25, which is /() 0 0 - 2 5 1 0 0 20 0 1 0 -14 0 0 1 4 / Thus (0(f))2 is the only elementary divisor of D, and it has multiplicity 1. For the cyclic generator, it suffices to find a function g in V for which 0(D) (o) ^ 0. * Sec. 7.4 The Rational Canonical Form 541 Since 0(^4)(e3) ^ 0, it follows that 0(D)(a;excos2a:) ^ 0; therefore g(x) = xex cos 2x can be chosen as the cyclic generator. Hence 0g = {xex cos 2x, D(xex cos 2x), D2(xex cos 2x), D3(xex cos 2x)} is a rational canonical basis for D. Notice that the function h defined by h(x) = xex sin 2x can be chosen in place of g. This shows that the rational canonical basis is not unique. It is convenient to refer to the rational canonical form and elementary divisors of a matrix, which are defined in the obvious way. Definitions. Let A G Mn x n(F) . The rational canonical form of A is defined to be the rational canonical form of LA- Likewise, for A, the elementary divisors and their multiplicities are the same as those of LA Let A be an n x n matrix, let C be a rational canonical form of A, and let 0 be the appropriate rational canonical basis for L^. Then C = [LAP, and therefore A is similar to C. In fact, if Q is the matrix whose columns are the vectors of 0 in the same order, then Q~XAQ = C. Example 5 For the following real matrix A, we find the rational canonical form C of A and a matrix Q such that Q~lAQ C. A = V 0 1 1 1 1 2 0 - 2 0 0 1 - 2 1 - 4 3 - 6 0 - 3 - 1 - 3 2 2 2 2 *) The characteristic polynomial of A is /(f) = (f2 + 2)2(f 2); therefore 01 (f) = f2 + 2 and 02(f) = f 2 are the distinct irreducible monic divisors of /(f). By Theorem 7.23, dim(K0,) = 4 and dim(K02) = 1. Since the degree of 0i (f) is 2, the total number of dots in the dot diagram of 0i(f) is 4/2 = 2, and the number of dots n in the first row is given by n = 4[dim(R5) - rank(0i (A))] = i [ 5 - r ank( ,4 2 + 2/)] = | [ 5 - 1 ] = 2 . Thus the dot diagram of 0i (f) is 542 Chap. 7 Canonical Forms and each column contributes the companion matrix 0 - 2 1 0 for 0i (f) = f2 + 2 to the rational canonical form C. Consequently 0i(f) is an elementary divisor with multiplicity 2. Since dim(K0;2) = 1, the dot diagram of 02(f) = f 2 consists of a single dot, which contributes the l x l matrix (2). Hence 02(f) is an elementary divisor with multiplicity 1. Therefore the rational canonical form C is C = / 0 1 0 0 V o - 2 0 0 0 0 0 0 0 1 0 0 0 - 2 0 0 0 0 0 0 2 / We can infer from the dot diagram of 0i(f) that if 0 is a rational canonical basis for L^, then 0D K^, is the union of two cyclic bases 0Vx and 0V2, where v and V2 each have annihilator 0i(f). It follows that both v and ?;2 lie in N(0i(L/t)). It can be shown that ((1 0 o 0 I w AA l 0 0 w (o 0 2 1 W ( 0 - l 0 X V > _ is a basis for N(0i(L,4)). Setting v e,, we sec that Av fi) 1 1 1 V Next choose v2 in K^, = N(0(Lyt)), but not in the span of 0Vi = {v,Av}. For example, V2 = e2. Then it can be seen that Av-? = / 2 -2 0 - 2 V-4/ and 0Vl U 0V2 is a basis for K0] . Sec. 7.4 The Rational Canonical Form 543 Since the dot diagram of 02(f) = f 2 consists of a single dot, any nonzero vector in K02 is an eigenvector of A corresponding to the eigenvalue A = 2. For example, choose M I V3= 1 1 w By Theorem 7.23, 0 = {vi, Av,V2, Av2,v%} is a rational canonical basis for LA- SO setting Q = (1 0 0 0 o 0 1 1 1 1 0 1 0 0 0 2 - 2 0 - 2 - 4 0 1 1 1 2/ we have Q~lAQ = C. Example 6 For the following matrix A, we find the rational canonical form C and a matrix Q such that Q~lAQ = C: A = (2 1 0 0 0 2 1 0 0 0 2 0 0 0 0 2 / Since the characteristic polynomial of A is /(f) = (f 2)4, the only irreducible monic divisor of /(f) is 0(f) f 2, and so K0 = R4. In this case, 0(f) has degree 1; hence in applying Theorem 7.24 to compute the dot diagram for 0(f), we obtain n = 4 - rank(0(A)) = 4 - 2 = 2, r2 = rank(0(^)) - rank((0(,4))2) = 2 - 1 = 1, and r3 = rank((0(4))2) - rank((0(^))3) = 1 - 0 = 1, where r^ is the number of dots in the ith row of the dot diagram. Since there are dim(R4) = 4 dots in the diagram, we may terminate these computations 544 Chap. 7 Canonical Forms with r$. Thus the dot diagram for A is Since (f 2) has the companion matrix '0 0 8' 1 0 -12 J) 1 6, and (f 2) has the companion matrix (2), the rational canonical form of A is given by C = Next we find a rational canonical basis for L.4. The; preceding dot diagram indicates that there are two vectors V and v2 in R4 with annihilators (0(f))3 and 0(f), respectively, and such that 0 = [0Vl U 0Vi} = {v., A;,, A2vi, v2} is a rational canonical basis for L.. Furthermore, Vj N((L4 - 2I)2), and v2 G N(L4 - 21). It can easily be shown that / 0 0 1 0 0 1 0 0 8 -12 6 0 0 0 0 2 J and N(L 4 -2 l )=span ({c 1 , e 4 } ) N((L A -2 l ) 2 ) = span({ei,e2,e4}). The standard vector C3 meets the criteria for v so we set V = ('3. It follows that Avi = Next we choose a vector v2 G N(L/i -21) that is not in the span of 0Vi. Clearly. v2 = < 1 satisfies this condition. Thus 0 2 V and A V{ = n 4 4 W r / o 0 1 I w > (o 1 2 W (i 1 4 w (0 0 0 W Sec. 7.4 The Rational Canonical Form 545 is a rational canonical basis for LA-Finally, let Q be the matrix whose columns are the vectors of 0 in the same order: Q = ft) 0 1 0 0 1 4 0 1 2 4 0 0 0 0 1/ T h e n C = Q - M Q . Direct Sums* The next theorem is a simple consequence of Theorem 7.23. Theorem 7.25 (Primary Decomposition Theorem). Let T be a linear operator on an n-dimensional vector space V with characteristic poly-nomial /(f) = ( - i r (0, ( o r ( 0 2 ( o r (0fc(f)r , where the 0;(f)'s (1 < % < k) are distinct irreducible monic polynomials and the ni 's are positive integers. Then the following statements arc true. (a) V = K^ K^2 K ^ . (b) If Ti (1 < i < k) is the restriction ofT to K ,^ and Cj is the rational canonical form of Tj, fhen C C2 Ck is the rational canonical form ofT. Proof. Exercise. 1 The next theorem is a simple consequence of Theorem 7.17. Theorem 7.26. Let T be a linear operator on a finite-dimensional vector space V. Then V is a direct sum of T-cyclic subspaces CVi, where each V{ lies in K^ for some irreducible monic divisor 0(f) of the characteristic polynomial ofT. Proof. Exercise. 1 EXERCISES 1. Label the following statements as true or false. (a) Every rational canonical basis for a linear operator T is the union of T-cyclic bases. 546 Chap. 7 Canonical Forms 3. (b) (c) (d) (e) (f) (g) If a basis is the union of T-cyclic bases for a linear operator T, then it is a rational canonical basis for T. There exist square matrices having no rational canonical form. A square matrix is similar to its rational canonical form. For any linear operator T on a finite-dimensional vector space, any irreducible factor of the characteristic polynomial of T divides the minimal polynomial of T. Let 0(f) be an irreducible monic divisor of the characteristic poly-nomial of a linear operator T. The dots in the diagram used to compute the rational canonical form of the restriction of T to K0 are in one-to-one correspondence with the vectors in a basis for K^. If a matrix has a Jordan canonical form, then its Jordan canonical form and rational canonical form are similar. 2. For each of the following matrices A G M n x(F) , find the rational canonical form C of A and a matrix Q G M n X n (F) such that Q~l AQ = C. (a) A = (c) A = (d) A = (e) A = F = R (b) A = 0 - 1 1 - 1 F = R F = R F = R For each of the following linear operators T, find the elementary divisors, the rational canonical form C, and a rational canonical basis Q. (a) T is the linear operator on P$(R) defined by T(f(x)) = f(0)x-f'(l). (b) Let S {sinx,cosa:,xsinx,xcos3;}, a subset of F(R, R), and let V = span(5). Define T to be the linear operator on V such that T(/) = / ' (c) T is the linear operator on M2x2(.R) defined by Sec. 7.4 The Rational Canonical Form 547 6. T(A) = 0 1 - 1 1 A. (d) Let S = {sin x sin?/, sin x cost/, cos x sin y, cos a: cost/}, a subset of F(R x R,R), and let V = span(S'). Define T to be the linear operator on V such that T(f)(x,y) = df(x,y) , df(x,y) dx dy 4. Let T be a linear operator on a finite-dimensional vector space V with minimal polynomial (0(f))m for some positive integer m. (a) Prove that R(0(T)) C N((0(T))m-1). (b) Give an example to show that the subspaces in (a) need not be equal. (c) Prove that the minimal polynomial of the restriction of T to R(0(T)) equals (0(f))m~1. 5. Let T be a linear operator on a finite-dimensional vector space. Prove that the rational canonical form of T is a diagonal matrix if and only if T is diagonalizable. Let T be a linear operator on a finite-dimensional vector space V with characteristic polynomial /(f) = (l)n0i(f)02(f), where 0i(f) and 02(f) are distinct irreducible monic polynomials and n dim(V). (a) Prove that there exist V,V2 G V such that v has T-annihilator i(t), ^2 has T-annihilator 02(f), and 0Vl U0V2 is a basis for V. Prove that there is a vector t^ G V with T-annihilator 0i(f)02(f) (b) (c) such that 0Va is a basis for V. Describe the difference between the matrix representation of T with respect to 0Vl U 0V2 and the matrix representation of T with respect to 0Va. Thus, to assure the uniqueness of the rational canonical form, we re-quire that the generators of the T-cyclic bases that constitute a rational canonical basis have T-annihilators equal to powers of irreducible monic factors of the characteristic polynomial of T. 7. Let T be a linear operator on a finite-dimensional vector space with minimal polynomial m = (Mt))mi(Mt)y w)r where the 0*(f)'s are distinct irreducible monic factors of /(f). Prove that for each i, m-i is the number of entries in the first column of the dot diagram for 0i(f). 548 Chap. 7 Canonical Forms 8. Let T be a linear operator on a finite-dimensional vector space V. Prove that for any irreducible polynomial 0(f), if 0(T) is not one-to-one, then 0(f) divides the characteristic polynomial of T. Hint: Apply Exercise 15 of Section 7.3. 9. Let V be a vector space and 0i,02,- ,0k be disjoint subsets of V whose union is a basis for V. Now suppose that 7i, 7 2 , . . , 7fc are linearly independent subsets of V such that span(7$) = span(/?) for all i. Prove that 71 U 72 U U 7fc is also a basis for V. 10. Let T be a linear operator on a finite-dimensional vector space, and suppose that 0(f) is an irreducible monic factor of the characteristic polynomial of T. Prove; that if 0(f) is the T-annihilator of vectors x and y, then x G Cy if and only if Cx = C. Exercises 11 and 12 arc concerned with direct sums. 11. Prove Theorem 7.25. 12. Prove Theorem 7.26. INDEX OF DEFINITIONS FOR CHAPTER 7 Companion matrix 526 Cycle of generalized eigenvectors 488 Cyclic basis 525 Dot diagram for Jordan canonical form 498 Dot diagram lor rational canonical form 535 Elementary divisor of a linear oper-ator 539 Elementary divisor of a matrix 541 End vector of a cycle 488 Generalized eigenspace 484 Generalized eigenvector 484 Generator of a cyclic basis 525 Initial vector of a cycle 488 Jordan block 483 Jordan canonical basis 483 Jordan canonical form of a linear op-erator 483 Jordan canonical form of a matrix 491 Length of a cycle 488 Minimal polynomial of a linear oper-ator 516 Minimal polynomial of a matrix 517 Multiplicity of an elementary divisor 539 llational canonical basis of a linear operator 526 Rational canonical form for a linear operator 526 Rational canonical form of a matrix 541 A p p e n d i c e s APPENDIX A SETS A set is a collection of objects, called elements of the set. If x is an element of the set A, then we write x E A; otherwise, we write x G' A. For example, if Z is the set of integers, then 3 G Z and i ^ Z. One set that appears frequently is the set of real numbers, which we denote by R throughout this text. Two sets A and B are called equal, written A = B,if they contain exactly the same elements. Sets may be described in one of two ways: 1. By listing the elements of the set between set braces { }. 2. By describing the elements of the set in terms of some characteristic property. For example, the set consisting of the elements 1, 2, 3, and 4 can be written as {1,2,3,4} or as {x: x is a positive integer less than 5}. Note that the order in which the elements of a set are listed is immaterial; hence {1,2,3,4} = {3,1,2,4} = {1,3,1,4,2}. Example 1 Let A denote the set of real numbers between 1 and 2. Then A may be written as A = {xeR:l550 Appendices The intersection of two sets A and B, denoted Ad B, is the set of elements that are in both A and B; that is, A fl B = {x: x G A and :r G B}. Two sets are called disjoint if their intersection equals the empty set. Example 2 Let A= {1,3.5} and B = {1,5,7,8}. Then AuB = {1,3,5,7,8} and A n J? = {1,5}. Likewise, if X = {1,2,8} and Y = {3,4,5}, then XUY = {1,2,3,4,5,8} and X (lY = 0. Thus X and Y are disjoint sets. The union and intersection of more than two sets can be defined analo-gously. Specifically, if Aj, A 2 , . . . , An are sets, then the union and intersec-tions of these sets are defined, respectively, by n M Ai = {x : x G Ai for some i 1 ,2 , . . . ,n} 2=1 and p | Ai = {rr: .r G A{ for all i = 1 ,2 , . . . , n}. j=i Similarly, if A is an index set and {Aa: a G A} is a collection of sets, the union and intersection of these sets arc defined, respectively, by M Att = {x: X E AQ for some a G A} r*6A and p | Aa = {x: x G An for all a G A}. aeA Example 3 Let A = {a R: a > 1}, and let 4 a = < x e R: < x < 1 + a I for each a G A. Then ( J A a = { G R: x > - 1 } and f) Aa = {x e R: 0 < x Appendix B Functions 551 By a relation on a set A, we mean a rule for determining whether or not, for any elements x and y in A, x stands in a given relationship to y. More precisely, a relation on A is a set S of ordered pairs of elements of A such that (x, y) G S if and only if x stands in the given relationship to y. On the set of real numbers, for instance, 'is equal to,' 'is less than,' and 'is greater than or equal to' are familiar relations. If S is a relation on a set A, we often write x ~ y in place of (x, y) G S. A relation 5 on a set A is called an equivalence relation on A if the following three conditions hold: 1. For each x G A, x ~ x (refcxivity). 2. If x ~ y, then y ~ x (symmetry). 3. If x ~ y and y ^ z, then x ~ 2 (transitivity). For example, if we define x ~ y to mean that x ?/ is divisible by a fixed integer n, then ~ is an equivalence; relation on the set of integers. APPENDIX B FUNCTIONS If A and B are sets, then a function / from A to B, written / : A * B, is a rule that associates to each element, x in A a unique element denoted f(x) in Z?. The element f(x) is called the image of x (under / ) , and x is called a preimage of f(x) (under / ) . If / : A B, then A is called the domain of / , B is called the codomain of / , and the set {fix): x E A} is called the range of / . Note that the range of / is a subset of B. If S C A, we denote by f(S) the set {/(x): x S} of all images of elements of S. Likewise, if T C B,we denote by f~l(T) the set {x G A: / (x) G T7} of all preimages of elements in T. Finally, two functions f: A > B and 552 Appendices Let / : A * B be a function and S Q A. Then a function / C called the composite of o and / . Thus (g o /)(x) = g(f(x)) for all x G A. For example, let A = B = C = R, f(x) = sinx. and .o(x) = x2 + 3. Then (g o / ) (x) = (.* Appendix C Fields 553 Definitions. A field F is a set on which two operations + and (called addition and multiplication, respectively) are defined so that, for each pair of elements x, y in F, there are unique elements x + y and x-y in F for which the following conditions hold for all elements a,b,c in F. ( F l ) a + b = b + a and a*6 = b-a (commutativity of addition and multiplication) (F2) (a + b)+c = a + (b + c) and (a-b)>c = a>(b-c) (associativity of addition and multiplication) (F 3) There exist distinct elements 0 and 1 in F such that 0 + a = a and 1 -a = a (existence of identity elements for addition and multiplication) (F 4) For each clement a in F and each nonzero clement b in F, there exist elements c and d in F such that a + c = 0 and tW = 1 (existence of inverses for addition and multiplication) (F 5) a-(b + c) =ci-b + a-c (distributivity of multiplication over addition) The elements x + y and x-y are called the sum and product, respectively, of x and y. The elements 0 (read 'zero') and 1 (read 'one') mentioned in (F 3) are called identity elements for addition and multiplication, respec-tively, and the elements c and d referred to in (F 4) are called an additive inverse for a and a multiplicative inverse for b, respectively. Example 1 The set of real numbers R with the usual definitions of addition and multi-plication is a field. Example 2 The set of rational numbers with the usual definitions of addition and multi-plication is a field. Example 3 The set of all real numbers of the form a + by/2, where a and b an; rational numbers, with addition and multiplication as in R is a field. Example 4 The field Z2 consists of two elements 0 and 1 with the operations of addition and multiplication defined by the equations 0 + 0 = 0, 0 + 1 = 1 + 0 = 1 , 1 + 1 = 0 , 0-0 = 0, 0-l = l -0 = 0. and 1-1 = 1. 554 Appendices Example 5 Neither the set of positive integers nor the set of integers with the usual definitions of addition and multiplication is a field, for in either case (F 4) does not hold. The identity and inverse elements guaranteed by (F 3) and (F 4) are unique; this is a consequence of the following theorem. Theorem C. l (Cancellation Laws). For arbitrary elements a, b, and c in a held, the following statements are true. (a) If a + b = c + b, then a = c. (b) If a-b = cb and b ^ 0, then a = c. Proof, (a) The proof of (a) is left as an exercise. (b) If b T& 0, then (F 4) guarantees the existence of an element d in the field such that b-d = 1. Multiply both sides of the equality a-b = c - b by d to obtain (a-6)-d = (c-fe)-d. Consider the left side of this equality: By (F 2) and (F 3), we have (a b) d = a (6- d) = a 1 = a. Similarly, the right side of the equality reduces to c. Thus a = c. I Corollary. The elements 0 and 1 mentioned in (F 3), and the elements c and d mentioned in (F 4), are unique. Proof. Suppose that 0' G F satisfies 0' + o, = a for each a G F. Since 0 + a = a for each a G F, we have 0' + a = 0 + a for each a G F. Thus 0' = 0 by Theorem C.l. The proofs of the remaining parts are similar. I Thus each element b in a field has a unique additive inverse and, if b ^ 0, a unique multiplicative inverse. (It is shown in the corollary to Theorem C.2 that 0 has no multiplicative inverse.) The additive inverse and the multi-plicative inverse of b are denoted by b and ft'1, respectively. Note that - ( -&) = 6 and (&-1)'1 = 6. Subtraction and division can be defined in terms of addition and multi-plication by using the additive and multiplicative inverses. Specifically, sub-traction of b is defined to be addition of b and division by b ^ 0 is defined to be multiplication by o_ 1; that is, a b = a + (6) and 1 - = d'b . b In particular, the symbol f denotes b l. Division by zero is undefined, but. o with this exception, the sum, product, difference, and quotient of any two elements of a field are defined. '*^B Appendix C Fields 555 Many of the familiar properties of multiplication of real numbers are true in any field, as the next theorem shows. Theorem C.2. Let a and b be arbitrary elements of a field. Then each of the following statements are true. (a) a-0 = 0. (b) ( - a ) -6 = a - ( -6 ) = - (a-6) . (c) ( - a ) - ( - 6 ) = a-6. Proof, (a) Since 0 + 0 = 0, (F 5) shows that 0 + a-0 = a-0 = a-(0 + 0) = a-0 + a-0. Thus 0 = a-0 by Theorem C.l. (b) By definition, (a-6) is the unique element of F with the property a-6 + [-(a-6)] = 0. So in order to prove that (a)-6 = (a-6), it suffices to show that a-6 + (a)-6 = 0. But a is the element of F such that a + ( - a ) = 0; so a-6 + ( - a ) - b= [a + (-a)] -6 = 0-6 = 6-0 = 0 by (F 5) and (a). Thus ( -a ) -6 = - (a-6) . The proof that a - ( -6) = - (a-6) is similar. (c) By applying (b) twice, we find that (-a)-(-b) = - M - 6 ) ] = - [ - (a -6) ] = a-6. | Corollary. The additive identity of a field has no multiplicative inverse. In an arbitrary field F, it may happen that a sum 1 + 1 H + 1 (p sum-mands) equals 0 for some positive integer p. For example, in the field Z2 (defined in Example 4), 1 + 1 = 0. In this case, the smallest positive integer p for which a sum of p l 's equals 0 is called the characteristic of F; if no such positive integer exists, then F is said to have characteristic zero. Thus Z2 has characteristic two, and R has characteristic zero. Observe that if F is a field of characteristic p / 0, then x + x + - + x (p summands) equals 0 for all x G F. In a field having nonzero characteristic (especially characteristic two), many unnatural problems arise. For this reason, some of the results about vector spaces stated in this book require that the field over which the vector space is defined be of characteristic zero (or, at least, of some characteristic other than two). Finally, note that in other sections of this book, the product of two ele-ments a and 6 in a field is usually denoted a6 rather than a 6. 556 Appendices APPENDIX D COMPLEX NUMBERS For the purposes of algebra, the field of real numbers is not sufficient, for there are polynomials of nonzero degree with real coefficients that have no zeros in the field of real numbers (for example, x2 + 1). It is often desirable to have a field in which any polynomial of nonzero degree with coefficients from that field has a zero in that field. It is possible to 'enlarge' the field of real numbers to obtain such a field. Definitions. A complex number is an expression of the form z = a+bi, where a and 6 are reai numbers called the real part and the imaginary part of z, respectively. The sum and product of two complex numbers z = a + bi and w = c+di (where a, b, c, and d are real numbers) are defined, respectively, as follows: z + w = (a + bi) + (c + di) = (a + c) + (6 + d)i and zw = (a + bi)(c + di) (ac bd) + (6c + ad)i. Example 1 The sum and product of z = 3 5i and w = 9 + 7% are, respectively, z + w = (3 - 5i) + (9 + 7i) = (3 + 9) + [(-5) + 7}i = 12 + 2i and zw = (3 - 5i)(9 + 7i) = [3-9 - (-5)-7] + [(-5)-9 + 3-7}i = 62 - 24i. Any real number c may be regarded as a complex number by identifying c with the complex number c + Oi. Observe that this correspondence preserves sums and products; that is, (c + Oi) + (d + Oi) = (c + d) + Qi and (c + 0i)(d + 0i) = cd + Oi. Any complex number of the form bi = 0 + bi, where 6 is a nonzero real number, is called imaginary. The product of two imaginary numbers is real since (bi)(di) = (0 + 6i)(0 + di) = (0 - 6d) + (6-0 + 0-d)i = -6d. In particular, for i = 0 + li, we have i i = 1. The observation that i2 = i i = 1 provides an easy way to remember the definition of multiplication of complex numbers: simply multiply two complex numbers as you would any two algebraic expressions, and replace i2 by 1. Example 2 illustrates this technique. Appendix D Complex Numbers Example 2 The product of 5 + 2i and 1 3i is ( - 5 + 2i)(l - Si) = -5(1 - Si) + 2i(l - Zi) = - 5 + 15i + 2i - 6i2 = 5 + 15i + 2 6 ( - l ) = 1 + 17i 557 Z The real number 0, regarded as a complex number, is an additive identity element for the complex numbers since (a + bi) + 0 = (a + bi) + (0 + Oi) = (a + 0) + (6 + 0)i = a + bi. Likewise the real number 1, regarded as a complex number, is a multiplicative identity element for the set of complex numbers since (a + 6i)-l = (a + bi)(l+0i) = (a-1 - 6-0) + (6-1 + a-0)i = a + bi. Every complex number a + bi has an additive inverse, namely (a) + (b)i. But also each complex number except 0 has a multiplicative inverse. In fact, (a + ) = a2 + b2 b2 In view of the preceding statements, the following result is not surprising. Theorem D . l . The set of complex numbers with the operations of addi-tion and multiplication previously defined is a Held. Proof. Exercise. 1 Definition. The (complex) conjugate of a complex number a + bi is the complex number a bi. We denote the conjugate of the complex number z byz. Example 3 The conjugates of 3 + 2i, 4 7i, and 6 are, respectively, -3 + 2i = -3-2i, 4-7i = 4 + 7i, and 6 = 6 + Oi = 6 - Oi = 6. The next theorem contains some important properties of the conjugate of a complex number. Theorem D.2. Let z and w be complex numbers. Then the following statements are true. 558 Appendices (a) z = z. (b) (z + w) = z + w. (c) ZW ~Z'W. (d) (-) = = ifw ^ 0. wJ w (e) z is a real number if and only if z = z. Proof. We leave the proofs of (a), (d), and (e) to the reader, (b) Let z a + bi and w c + di, where a, 6, c, d G R. Then [z + w) = (a + c) + (6 + d)i = (a + c) - (6 + d)i = (a bi) + (c di) = z + w. (c) For z and to, we have zw = (a + bi)(c + di) = (ac bd) + (ad + bc)i (ac 6d) (ad + bc)i (a 6i)(c di) = z-w. 13 For any complex number z = a + bi, z~z is real and nonnegative, for z~z (a + bi)(a bi) = a2 + b2. This fact can be used to define the absolute value of a complex number. Definition. Let z = a + bi, where a,b G R. The absolute value (or modulus) of z is the real number /a2 + 62. We denote the absolute value of z by z. Observe that zz = |z|2. The fact that the product of a complex number and its conjugate is real provides an easy method for determining the quotient of two complex numbers; for if c + di ^ 0, then a + 6i a + 6i c di (ac + bd) + (be ad)i ac + bd, be ad . c2+d2 + d? r/'2 c + di c + di c di Example 4 To illustrate this procedure, we compute the quotient (1 + 4i)/(3 2i): l + 4 i _ l + 4 i 3 + 2 f _ - 5 + 14i_ 5 14. 3 - 2i ~ 3 - 2 i ' 3 + 2i ' 9 + 4 13 + 13*' The absolute value of a complex number has the familiar properties of the absolute value of a real number, as the following result shows. Theorem D.3. Let z and w denote any two complex numbers. Then the following statements are true. B~ Appendix D Complex Numbers (a) zw = z-w. 559 (b) - = 1 1 ifw^O. w w (c) z + w < z + w. (d) z - w < z + w. Proof, (a) By Theorem D.2, we have zw2 = izw)izw) = (zw)(z w) = (zz)(ww) |2|2|w;|2, proving (a). (b) For the proof of (b), apply (a) to the product ( ) w. w/ (c) For any complex number x = a + 6i, where a, 6 G R, observe that x + x = (a + 6i) + (a - bi) = 2a < 2/a2 + b2 = 2|x|. Thus x + x is real and satisfies the inequality x + x < 2|x|. Taking x = wz, we have, by Theorem D.2 and (a), wz + wz < 2wz = 2|w;||2| = 2|z||tu|. Using Theorem D.2 again gives z + w = iz + w)(z + w) = (z + w)(z + w) = zz + wz + ZW + WW < zf + 2|z|H + wf = (|z| By taking square roots, we obtain (c). (d) From (a) and (c), it follows that ' z = 1(2 + w) w < z + w + I w = z + w + w. So z |t^| < z + w, proving (d). 1 It is interesting as well as useful that complex numbers have both a ge-ometric and an algebraic representation. Suppose that z a + bi, where a and 6 are real numbers. We may represent z as a vector in the complex plane (see Figure D.l(a)). Notice that, as in R2, there are two axes, the real axis and the imaginary axis. The real and imaginary parts of z are the first and second coordinates, and the absolute value of z gives the length of the vector z. It is clear that addition of complex numbers may be represented as in R2 using the parallelogram law. 560 Appendices imaginary axis (b) Figure D.l In Section 2.7 (p.132). we introduce Filler's formula. The special case (,tn _ c o s g _|_ j g m 0 js 0 particular interest. Because of the geometry we have introduced, we may represent the vector c' as in Figure D.l(b); that is, el is the unit vector that makes an angle 6 with the positive real axis. From this figure, we sec1 that any nonzero complex number z may be depicted as a multiple of a unit vector, namely, z = |^|e*^, where 0 is the angle that the vector z makes with the positive real axis. Thus multiplication, as well as addition, has a simple geometric interpretation: If z = ze and w = we,w are two nonzero complex numbers, then from the properties established in Section 2.7 and Theorem D.3. we have .// >o we' ,i(, and makes the angle 0 + uj with the positive real axis. Our motivation for enlarging the set of real numbers to the set of complex numbers is to obtain a field such that every polynomial with nonzero degree having coefficients in that field has a zero. Our next result guarantees that the field of complex numbers has this property. Theorem D.4 (The Fundamental Theorem of Algebra). Suppose that p(z) = o.z' + a iz' l + + az + a{) is a polynomial in P(C) of degree n > 1. Then />(.r) has a zero. The following proof is based on one in the book Principles of Mathematical Analysis 3d., by Walter Rudin (McGraw-Hill Higher Education. New York, 1976). Proof. We want to find zrj in C such that p(zo) = 0. Let m be the greatest lower bound of {|/>(c)|: 2 G ('}. For z = s > 0. we have p(z) -= anzn +a_ i .-i an 7 s Appendix D Complex Numbers 561 > |an | |2n | - lon-iU^p'1 |a0 | = ansn - | a n _ i | s n _ 1 |ao| = sn[an - | a n _i | s _ 1 a0s~n]. Because the last expression approaches infinity as s approaches infinity, we may choose a closed disk D about the origin such that p(z) > m + 1 if z is not in D. It follows that rn is the greatest lower bound of {|p(z)|: z G D). Because D is closed and bounded and p(z) is continuous, there exists ZQ in D such that |p(zo)| = m. We want to show that rn = 0. We argue by contradiction. Assume that rn ^ 0. Let q(z) = :-. Then q(z) is a polynomial of P(zo) degree n, q(0) = 1, and q(z) > 1 for all z in C. So we may write q(z) = 1 + bkzk + bk+1zh+1 + + bnzn, where 6fc 7^ 0. Because has modulus one. we may pick a real number 9 bk such that etkd = ~-, or etkdbk = l&J. For any r > 0, we have bk q(rei9) = 1 + bkrkelk6 + &fc+1r*+1e*+1>' + + bnrnein6 = 1 - bkrk + bk+1rk+1ei(k+l)d + + bnrnein0. Choose r small enough so that 1 |6fc[rfc > 0. Then q(rei9) < 1 - |6fc|rfc + |6fc+i|rfc+1 + + |6n | rn = l - r f c [ |6 fc | - |6 f c + i | r |6n |r'-*]. Now choose r even smaller, if necessary, so that the expression within the brackets is positive. We obtain that q(rel0) < 1. But this is a contradiction. I The following important corollary is a consequence of Theorem D.4 and the division algorithm for polynomials (Theorem E.l). Corollary. If p(z) = anzn + an-izn~l + + az + ao is a polynomial of degree n > 1 with complex coefficients, then there exist complex numbers Ci, c2, , cn (not necessarily distinct) such that p(z) = an(z - cA)(z - c2) (z - Cn). Proof. Exercise. | A field is called algebraically closed if it has the property that every polynomial of positive degree with coefficients from that field factors as a product of polynomials of degree 1. Thus the preceding corollary asserts that the field of complex numbers is algebraically closed. 562 APPENDIX E POLYNOMIALS Appendices In this appendix, we discuss some useful properties of the polynomials with coefficients from a field. For the definition of a polynomial, refer to Sec-tion 1.2. Throughout this appendix, we assume that all polynomials have coefficients from a fixed field F. Definition. A polynomial f(x) divides a polynomial g(x) if there exists a polynomial q(x) such that g(x) = f(x)q(x). Our first result shows that the familiar long division process for polyno-mials with real coefficients is valid for polynomials with coefficients from an arbitrary field. Theorem E. l (The Division Algorithm for Polynomials). Let f(x) be a polynomial of degree n, and let g(x) be a polynomial of degree rn > 0. Then there exist unique polynomials q(.r) and r(x) such that f(x)=q(x)g(x) + r(x), where the degree of r(x) is less than m. (1) Proof. We begin by establishing the existence of q(x) and r(x) that sat-isfy (1). CASE 1. If n < m, take q(x) = 0 and r(x) = f(x) to satisfy (1). CASE 2. When 0 < rn < n, we apply mathematical induction on n. First suppose that n = 0. Then m = 0, and it follows that f(x) and g(x) are nonzero constants. Hence we may take q(x) f(x)/g(x) and r(x) = 0 to satisfy (1). Now suppose that the result is valid for all polynomials with degree less than n for some fixed n > 0, and assume that f(x) has degree n. Suppose that f{x) = anxn + an-ixn + aix + ao and g(x) = bmxm + bm-ix-1 + + bix + bo, and let h(x) be the polynomial defined by hi.r) = fix)-anb-l]-r'- Appendix E Polynomials 563 Combining (2) and (3) and solving for f(x) gives us f(x) = q(x)g(x) + r(x) with q(x) a n 6~ 1 x' _ m + 0 by mathematical induction. This establishes the existence of q(x) and r(x). We now show the uniqueness of q(x) and r(x). Suppose that Oi(x), g2(x), ri(x), and r2(x) exist such that n (x ) and r2(x) each has degree less than m and f(x) = oi(x)o(x) + ri(x) = q2(x)g(x) + r2(x). Then ki(x-) - q2(x)]g(x) = r2(x) - n(x). (4) The right side of (4) is a polynomial of degree less than m. Since g(x) has degree m, it must follow that q(x) q2(x) is the zero polynomial. Hence 564 Appendices For any polynomial f(x) with coefficients from a field F, an element a G F is called a zero of f(x) if f(a) = 0. With this terminology, the preceding corollary states that a is a zero of f(x) if and only if x a divides f(x). Corollary 2. Any polynomial of degree n > 1 has at most n distinct zeros. Proof. The proof is by mathematical induction on n. The result is obvious if n 1. Now suppose that the result is true for some positive integer n, and let f(x) be a polynomial of degree n + 1. If f(x) has no zeros, then there is nothing to prove. Otherwise, if a is a zero of / (x) , then by Corollary 1 we may write f(x) = (x a)o(x) for some polynomial q(x). Note that q(x) must be of degree n; therefore, by the induction hypothesis, q(.r) can have at most n distinct zeros. Since any zero of f(x) distinct from a is also a zero of q(x), it follows that / (x) can have at most n + 1 distinct, zeros. Polynomials having no common divisors arise naturally in the study of canonical forms. (See Chapter 7.) Definition. Two nonzero polynomials are called relatively prime if no polynomial of positive degree divides each of them. For example, the polynomials with real coefficients f(x) = x2(x - 1) and h(x) = (x l)(x 2) are not relatively prime because x 1 divides each of them. On the other hand, consider f(x) and g(x) = (x 2)(x 3), which do not appear to have common factors. Could other factorizations of f(x) and g(x) reveal a hidden common factor? We will soon see (Theorem E.9) that the preceding factors are the only ones. Thus f(x) and g(x) are relatively prime because they have; no common factors of positive degree. Theorem E.2. If fi(x) and f2(x) are relatively prime polynomials, there exist polynomials q(x) and q2ix) such that qi(x)fi(x) + q2(x)f2(x) = l, where 1 denotes the constant, polynomial with value 1. Proof. Without loss of generality, assume that the degree of fi(x) is greater than or equal to the degree of /2(x). The proof is by mathematical induction on the degree of /2(x). If /2(x) has degree 0, then /2(x) is a nonzero constant c. In this case, we can take q(x) = 0 and f/2(x) = 1/c. Now suppose that the theorem holds whenever the polynomial of lesser degree has degree less than n for some positive integer n. and suppose that /2(x) has degree n. By the division algorithm, there exist polynomials q(x) and r(x) such that r(x) has degree less than n and / , ( x ) = g ( x ) / 2 ( x ) + r(x). (5) Appendix E Polynomials 565 Since / i (x) and f2(x) are relatively prime, r(x) is not the zero polynomial. We claim that /2(x) and r(x) are relatively prime. Suppose otherwise; then there exists a polynomial g(x) of positive degree that divides both /2(x) and r(x). Hence, by (5), g(x) also divides / i (x) , contradicting the fact that / i (x) and /2(x) are relatively prime. Since r(x) has degree less than n, we may apply the induction hypothesis to /2(x) and r(x). Thus there exist polynomials ai (x) and p2 (x) such that 9i(x)f2(x)+g2(x)r(x) = l. (6) Combining (5) and (6), we have 1 = gi(x)f2(x) + g2(x) [fi(x) - q(x)f2(x) = g^(x)fi(x) + [oi(x) - g2(x)q(x)] f2(x). Thus, setting q(x) = p2(^) and 566 Appendices Example 2 Let T be the linear operator on R2 defined by T(a,6) = (2a + 6, a 6), and let f(x) = x2 + 2x - 3. It is easily checked that T2(a, 6) = (5a + 6, a + 26); so /(T)(a,6) = (T2 + 2T-3 l ) (a ,6 ) = (5a + 6. a + 26) + (4a + 26,2a - 26) - 3(a, 6) = (6a+ 36,3a-36) . Similarly, if then A = f(A) = A2+2A-3I = 1 2 + 2 1 - 1 2 l)-3 r l -1 0 1 6 3 3 - 3 The next three results use this notation. Theorem E.3. Let f(x) be a polynomial with coefficients from a field F, and let T be a linear operator on a vector space V over F. Then the following statements are true. (a) / (T) is a linear operator on V. (b) If8 is a finite ordered basis for V and A = [T)0, then [f(T)}0 = f(A). Proof. Exercise. 1 Theorem E.4. Let T be a linear operator on a vector space V over a field F, and let A be a square matrix with entries from F. Then, for any polynomials fi(x) and /2(x) with coefficients from F, (a) / i (T)/ 2 (T) = /2(T)/1(T) (b) /1(A)/2(A) = /2(A)/1(A). Proof. Exercise. 1 Theorem E.5. Let T be a linear operator on a vector space V over a field F, and let A be an n x n matrix with entries from F. If f (x) and /2(x) are relatively prime polynomials with entries from F, then there exist polynomials q(x) and q2(x) with entries from F such that (a) ai(T)/!(T)+o2(T)/2(T) = l (b) q1{A)f1(A) + q2{A)f2(A) = I. Proof. Exercise. w Appendix E Polynomials 567 In Chapters 5 and 7, we are concerned with determining when a linear operator T on a finite-dimensional vector space can be diagonalized and with finding a simple (canonical) representation of T. Both of these problems are affected by the factorization of a certain polynomial determined by T (the characteristic polynomial ofT). In this setting, particular types of polynomi-als play an important role. Definitions. A polynomial f(x) with coefficients from a field F is called monic if its leading coefficient is 1. If f(x) has positive degree and cannot be expressed as a product of polynomials with coefficients from F each having positive degree, then f(x) is called irreducible. Observe that whether a polynomial is irreducible depends on the field F from which its coefficients come. For example, f(x) = x2 + 1 is irreducible over the field of real numbers, but it is not irreducible over the field of complex numbers since x2 + 1 = (x + i)(x i). Clearly any polynomial of degree 1 is irreducible. Moreover, for polyno-mials with coefficients from an algebraically closed field, the polynomials of degree 1 are the only irreducible polynomials. The following facts are easily established. Theorem E.6. Let 0(x) and f(x) be polynomials. If 0(x) is irreducible and 4>(x) does not divide f(x), then 0(x) and f(x) are relatively prime. Proof. Exercise. i Theorem E.7. Any two distinct irreducible monic polynomials arc rela-tively prime. Proof. Exercise. I Theorem E.8. Let f(x), g(x), and 0(x) be polynomials. If 0(x) is ir-reducible and divides the product f(x)g(x), then 0(x) divides f(x) or (x) divides g(x). Proof. Suppose that 0(x) does not divide f(x). Then 0(x) and f(x) are relatively prime by Theorem E.6, and so there exist polynomials tfi(x) and q2(x) such that l = O l (x)0(x)+o 2 (x) / (x) . Multiplying both sides of this equation by g(x) yields g(x) = ai(x)0(x)p(x) + a2(x)/(x)o(x). (7) Since 0(x) divides f(x)g(x), there is a polynomial h(x) such that f(x)g(x) = 4>(x)h(x). Thus (7) becomes g(x) = qi(x)(f>(x)g(x) + c2(x)0(x)a(x) = 0(x) qi(x)g(x) + q2(x)h(x)}. So 0(x) divides g(x). 1 568 Appendices Corollary. Let 0(x), 0i (x), 0 2 (x ) , . . . , n(x) be irreducible monic polyno-mials. If 0(x) divides the product 0i(x)02(x) -0n(x), then 0(x) = 0i(x) for some i (i 1 ,2 , . . . , n). Proof. We prove the corollary by mathematical induction on n. For n = 1, the result is an immediate; consequence of Theorem E.7. Suppose then that for some n > 1, the corollary is true for any n 1 irreducible monie polynomials, and let 0i (x), 0 2 (x ) . . . . , 0n(x) be n irreducible polynomials. If 0(x) divides 0l(-r)02(.r) (t)n(x) = [0l(x)02(x) 0_i(x)] 0n(x), then 0(x) divides the product 0i(x)02(x) 0_i (x) or 0(x) divides 0n(x) by Theorem E.8. In the first case, 0(x) = 0i(x) for some i (i = 1 ,2 , . . . , n 1) by the induction hypothesis; in the second case, 0(x) = n(x) by Theorem E.7. We are now able to establish the unique factorization theorem, which is used throughout Chapters 5 and 7. This result states that every polynomial of positive; degree is uniquely expressible as a. constant times a product of irreducible monic polynomials. Theorem E.9 (Unique Factorization Theorem for Polynomials). For any polynomial fix) of positive degree, there exist a unique constant c; unique distinct irreducible monic polynomials 0i(.r). 0 2 (x ) , . . . . 4>k(x); and unique positive integers n, n 2 , . . . , nk such that f(x) = Appendix E Polynomials 569 guarantees that both g(x) and h(x) factor as products of a constant and pow-ers of distinct irreducible monic polynomials. Consequently f(x) g(x)h(x) also factors in this way. Thus, in either case, f(x) can be factored as a product of a constant and powers of distinct irreducible monic polynomials. It remains to establish the uniqueness of such a factorization. Suppose that / ( x ) = c [ 0 i ( x ) r [0 2 (x )p [0 f c (x) r = d[Mx)}mi[Mx)}m2---lMx)]n (8) where c and d are constants, 0i(x) and i/)j(x) are irreducible monic polynomi-als, and ni and m,j are positive integers for i = 1 ,2 , . . . , k and j 1 ,2 , . . . , r. Clearly both c and d must be the leading coefficient of /(x); hence c = d. Dividing by c, we find that (8) becomes [0i ( x ) r [0 2 (x )p [0fc(x)r = [Mx)}1 [Mx)r> IM*)}1 (9) So 0i(x) divides the right side of (9) for i = 1 ,2 , . . . , k. Consequently, by the corollary to Theorem E.8, each 0i(x) equals some ipj(x), and similarly, each ij)j(x) equals some 0(x). We conclude that r = k and that, by renumbering if necessary, 0i(x) = i^i(x) for i = 1,2,... ,k. Suppose that n ^ m-j for some i. Without loss of generality, we may suppose that i = 1 and ni > mi. Then by canceling [0i(x)]mi from both sides of (9), we obtain [0 i (x)p-' [02(x)]'2 [0 f c (x) r = [ 0 2 ( x ) p [0fc(x)f (10) Since ni mi > 0, 0i(x) divides the left side of (10) and hence divides the right side also. So 0i(x) = 0i(x) for some i = 2 , . . . , k by the corollary to Theorem E.8. But this contradicts that 0 i (x) ,0 2 (x) , . . . ,0fc(x) are distinct. Hence the factorizations of f(x) in (8) are the same. 1 It is often useful to regard a polynomial f(x) = anxn H h aix + ao with coefficients from a field F as a function f:F*F. In this case, the value of / at c G F is f(c) = ancn + - aC + ao- Lmfortunately, for arbitrary fields there is not a one-to-one correspondence between polynomials and polynomial functions. For example, if f(x) = x2 and g(x) = x are two polynomials over the field Z2 (defined in Example 4 of Appendix C), then f(x) and g(x) have different degrees and hence are not equal as polynomials. But f(a) = g(a) for all a G Z2, so that / and g are equal polynomial functions. Our final result shows that this anomaly cannot occur over an infinite field. Theorem E.10. Let f(x) and g(x) be polynomials with coefficients from an infinite Geld F. If f(a) = g(a) for all a, G F, then f(x) and g(x) are equal. Proof. Suppose that / (a) = g(a) for all a G F. Define h(x) = f(x) g(x), and suppose that h(x) is of degree n > 1. It follows from Corollary 2 to 570 Appendices Theorem E.l that h(x) can have at most n zeroes. But h(a) = f(a)-g(a) = 0 for every a G F, contradicting the assumption that h(x) has positive degree. Thus h(x) is a constant polynomial, and since h(a) 0 for each a G F, it follows that h(x) is the zero polynomial. Hence f(x) = g(x). 1 A n s w e r s t o S e l e c t e d E x e r c i s e s C H A P T E R 1 SECTION 1.1 1. Only the pairs in (b) and (c) are parallel. 2. (a) x = (3, -2,4) + t(-8,9, -3) (c) x = (3, 7, 2) + t(0,0, -10) 3. (a) x = (2, -5, -1) + s(-2,9, 7) + t(-5,12,2) (c) x = (-8,2,0) + s(9,1,0) + t(14, -7,0) SECTION 1.2 1. (a) T (b) F (c) F (d) F (e) T (f) F (g )F ( h ) F ( i )T ( j )T ( k ) T 3. M13 = 3, M2i = 4, and M22 = 5 6 3 2 , , /8 20 -12N 4' ( a ) 1 -4 3 9 ( C ) U 0 28 (e) 2x4 + x3 + 2x2 - 2x + 10 (g) 10rc7 - 30x4 + 40x2 - 15x 13. No, (VS 4) fails. 14. Yes 15. No 17. No, (VS 5) fails. 22. 2mn SECTION 1.3 1. (a) F (b) F (c) T (d) F (e) T (f) F ( 2 , ( a ) ( _ 2 _ i ) i the trace i s - 5 (c) ( ' * J fj ( l (e) - 1 3 < 5y (g) (5 6 7) (g)F 8. (a) Yes (c) Yes (e) No 11. No, the set is not closed under addition. 15. Yes 571 572 Answers to Selected Exercises SECTION 1.4 1. (a) T (b) F (c) T (d) F (e) T (f) F 2. (a) {r( 1,1,0,0) + s(-3,0, -2,1) + (5,0,4,0) :r,sAnswers to Selected Exercises 573 2. The nullity is 1, and the rank is 2. T is not one-to-one but is onto. 4. The nullity is 4, and the rank is 2. T is neither one-to-one nor onto. 5. The nullity is 0, and the rank is 3. T is one-to-one but not onto. 10. T(2,3) = (5,11). T is one-to-one. 12. No. SECTION 2.2 1. (a) T (b) T (c) F (2 - 1 2. (a) 3 4 (c) (2 1 i 0 / /0 0 0 1 (f) 0 0 1 0 0 1 0 0 1 0 0 0/ / - * - 1 3. m j = 0 1 and [T]; V I o / / l 0 0 0 5. (a) 0 0 1 0 0 1 0 0 (b) 0 0 0 1/ / l 1 0 ON 10. 0 1 1 0 0 0 1 0 0 0 0 -- 1 0 0 0 1/ SECTION 2.3 1. (a) F (b) T (c) F (g) F (h) F (i) T 2. (a)A(2 + 3C7)=(2 ^ ,.v /23 19 0 /2 3 0 3. (a)[T]^= 0 3 6 , [U]J 0 0 4/ 4. (a) { l - 1 4 < 6y (c) (5) ( d ) T -3) ( g ) ( l (~ l = (0 1 2 2 0 0 0^ 0 ( d ) T (J)T i ) and / I = 0 V ( e ) T / 0 '. f ) F I 1 (d) - 1 4 5 V i o i / 0 - 0 1 3 3 2 3 2 *-; 3 3/ o 2 0 (e) 2/ ) / 1 - 2 0 V v (e) F (f) F and A(BD) CB = (27 7 1 0 0 1 , anc - 1 o) -() 9) /2 6 [UT]2 = 0 0 2 0 6 4 - 6 574 Answers to Selected Exercises 12. (a) No. (b) No. SECTION 2.4 1. (a) F ( g ) T 2. (a) No 3. (a) No 19. (a) [T]0 = SECTION 2.5 ( d ) F (d) No (d) No ( e ) T (e) No ( f )F (f) Yes 7. (a)T(x,y) = 1 + m 2 ((1 - m2)x + 2my, 2mx + (m2 - l)y) SECTION 2.6 1. (a) F (b) T (c) T (d) T (e) F (f) T (g) T (h) F 2. The functions in (a), (c), (e), and (f) are linear functionals. 3. (a) h(x,y,z) = x- y, h(x,y,z) = y, and f3ix,y,z) = -x + z 5. The basis for V is {pi(x),P2(x)}, where pi(x) = 2 - 2x and p?(x) = - 5 + x. 7. (a) T'(f) = g, where g(o + bx) = -3a - 46 w v f . = ( z i ; ) ( o [ T ] s = ( - ; -f) Answers to Selected Exercises SECTION 2.7 1. (a) T (b) T (c) F (d) F (e) T (f) F (g) T 2. (a) F (b) F (c) T (d) T (e) F 3. (a) {e-*,te-*} (c) {e'Ste-*,**,*^} (e) {e~*,e* cos 2t,e tsin2t} 575 4. ( a H e ' 1 4 - ^ 2 , ^ 1 - ^ / 2 } (c){ l ,e -4* -2t C H A P T E R 3 SECTION 3.1 1. (a) T (b) F (c) T (d) F (e) T (f) F (g) T (h) F (i) T 2. Adding 2 times column 1 to column 2 transforms A into B. /0 0 1 / l 0 0 3. (a) (0 1 0 (c) (0 1 0) 1 0 0/ 2 0 1/ (e) F (f) T - 3 -5. (a) The rank is 2, and the inverse is I . (c) The rank is 2, and so no inverse exists. (e) The rank is 3, and the inverse is (g) The rank is 4, and the inverse is 6. (a) T _ 1 (ax2 + bx + c) = -ax2 - (4a + b)x - (10a + 2b + c) (c) T'1(a,6,c) = (la-b+c,a-c,- + b+ c) (e) T~1(a,b,c) = (a-b+c)x2 + (-a+c)x + b ( 1 0 0 (1 0 0 / l 0 0 (1 2 0 / l 0 0 1 O i l 1 0 0 - 2 O l l O 1 O l i o 1 i o l / l o o i / o o 1/ o o 1/ Vo - l 0 1/ (1 0 1 10 1 0 Vo o i 576 Answers to Selected Exercises 20. (a) / 1 3 0 0 0 - 2 1 0 0 0 1 0 0 0 0 0 - 2 0 0 0 V 0 1 0 0 0/ SECTION 3.3 1. (a) F (b) F (c) T (d) F 2. (a) ( e )F ( f )F (g )T ( h ) F (e) 3. (a) (e) (g) (g) (~3 1 1 V . + t -3 ' t G R 1 ^ (1 0 0 w + r (~2 1 0 V oy1 + .s (3 0 1 w + t (-l 0 0 ^ 1/ : r, s, t G It 0 0 w + r 1 0 : r, .s, G /? 4. (b) (1) .4-1 = 6. T-1{(1,H)} = ^ 2 9 2 1 3 9/ ( 2 ) G -' 3N -2, + t - 1 : t G /? V 0/ 7. The systems in parts (b), (c), and (d) have solutions. 11. The farmer, tailor, and carpenter must have incomes in the proportions 4 : 3 : 4 . 13. There must be 7.8 units of the first commodity and 9.5 units of the second. SECTION 3.4 1. (a) F (b) T (c) T (d) T (e) F (f) T (g) T L Answers to Selected Exercises 577 2. (a) - 3 (c) (e) f [ /4 0 1 w (f H w + 8 /1 U 2 w ^ : r,s e R t (g) < / - 2 3 0 7 9 V J 4. (a) ^ 3 1 3 0 I w + t ( l - 1 1 z) : te R , (c) There are no solutions. / 1 0 2 1 4 5. ( - 1 - 1 3 - 2 - 7 V 3 1 1 0 - 9 / 7. {i,2,5} 11. (b) {(1,2,1,0,0), (2,1,0,0,0), (1,0,0,1,0), (-2,0,0,0,1)} 13. (b) {(1,0,1,1,1,0), (0,2,1,1,0,0), (1,1,1,0,0,0), (-3,-2,0,0,0,1)} C H A P T E R 4 SECTION 4.1 1. (a) F (b) T (c) F 2. (a) 30 (c) - 8 3. (a) -10 + 15i (c) -24 4. (a) 19 (c) 14 (d) F (e) T SECTION 4.2 1. (a) F (b) T 3. 42 5. -12 13. - 8 15. 0 ( c )T 7. -12 17. -49 ( d ) T 9. 22 19. -28 (e )F 11. - 3 -i 2 (f) F (g) F (h) T 21. 95 578 Answers to Selected Exercises SECTION 4.3 1. (a) F (b) T (c) F (d) T (e) F (f) T 3. (4,-3,0) 5. ( -20,-48,-8) 7. (0.-12. 16) (g) F (h) F 24. tn +ari-it' 2 6- ( - : (e) H +tti< + ao /10 0 (c) 0 -20 0 -An An -3 i 0 0 4 -1 + i 0 10+16i - 5 - 3 i 3 + 3i, 0 - 8 (g) 18 28 -20 -21 -6N 37 48 14 -16, SECTION 4.4 1. (a) T (b) T (c) T (d) F (e) F (g) T (h) K (i) T (j) T (k) T 2. (a) 22 (c) 2 - 4i 3. (a) -12 (c) -12 (e) 22 (g) - 3 4. (a) 0 (c) -49 (e) -28 - i (g) 95 SECTION 4.5 1. (a) F (b) T (c) T (d) F (e) F 3. No 5. Yes 7. Yes 9. No ( f ) T ( f )T C H A P T E R 5 SECTION 5.1 1. (a) F (b) T (c) T (g) F (h) T (i) T 2. (a) [Tk3=(_$ o) ' ' (e) [TJ (d) F (e) F (f) F (j) F (k) F (-1 0 0N (c) [T]3 = 0 1 0 V 0 0 -1, o 0 yes no - 1 1 0 0 - 1 1 0 0 - 1 0 0 0 0 -1) 3. (a) The eigenvalues are 4 and 1, a basis of eigenvectors is '2 ( l M ^ (2 L , ( 3 y v - v r ' vs - v an~ Answers to Selected Exercises 579 4. (a) A = 3,4 (b) A = -1,1,2 (f) A = 1,3 (h) A = -1,1,1,1 0 = (i) A = 1,1, - 1 , - 1 (3 = (j)A = - l , l , 5 (3 = 26. 4 SECTION 5.2 1. (a) F (b) F (c) F (g) T (h) T (i) F /? = {(3,5),(1,2)} /? = {(1,2,0),(1,-1,-1),(2,0,-1)} (3 = {-2 + x, -4 + x2, - 8 + x3, x} -1 0 /0 l / l 0 o 1/ ' o oy ' [o i 1 0 /0 l (-1 0 i o) ' vo iy ' v i o 0 A (1 0 /0 1 - l o j ' l o - l / ' l l 0 0 0 1 0 0 - 1 0 1 1 0 0 1 ( f )F (d) T (e) T 2. (a) Not diagonalizable (c) Q = I J / i i i (e) Not diagonalizable (g) Q = 2 - 1 0 V- l 0 -l) 3. (a) Not diagonalizable (c) Not diagonalizable (d) (3 = {x - x2,1 - x - x2, x + x2} (e) (3 = {(1,1), (1, -1)} 7 4 _ l / 5 n + 2 ( - i r 2 ( 5 ' ) - 2 ( - l ) ' 3 5 n - ( - l ) n 2(5)n + ( - l ) n 14. (b)x(t) = c 1 e 3 ^ - ^ + c 2 e - 2 ^ _ 1 1 ) ( c ) x ( t ) = e ' C I 0 + C 2 [ + c3e 2til SECTION 5.3 1. (a) T (b) T (c) F (d) F (e) T (g )T ( h ) F ( i )F ( j )T 2. (a) 0 0 0 0 (c) / - I 0 -1 (g) - 4 1 - 2 2 0 2 / 6. One month after arrival, 25% of the patients have recovered, 20% are ambu-latory, 41% are bedridden, and 14% have died. Eventually | | recover and | die. 580 Answers to Selected Exercises 7. f . 8. Only the matrices in (a) and (b) are regular transition matrices. 9. (a) (e) 10. (a) (c) (e) l 3 l 2 3 1 3 ) 0 0 V (c) No limit exists. (g) 0 0 l 2 '0.225' 0.441 ^0.334, '0.372^ 0.225 v0.403, '0.329N 0.334 ,0.337, after two stages and after two stages and '0.50N 0.20 .0.30, eventually eventually after two stages and l eventually 12. yjj new, JQ once-used, and j twice-used 13. In 1995, 24% will own large cars, 34% will own intermediate-sized cars, and 42% will own small cars; the corresponding eventual percentages are 10%, 30%, and 60%. 20. e = I and e7 = el. SECTION 5.4 1. (a) F (b) T (c) F (d) F (e) T (f) T 2. The subspaces in (a), (c), and (d) are T-invariant. A / i / i (g )T 6. (a) (c) 0 0 - 1 0 ' 1 ' 2 i w W V 2/ 9. (a) -tit2 - 3t + 3) (c) 1 - t 10. (a) tit - l)(t2 - 3t + 3) (c) (t - l)3(t '2 - 2 - 4 1 3 1) 1 18. (c) A'1 = I 0 z 0 0 - 2 Answers to Selected Exercises 31. (a) t2 - 6t + 6 (c) -(t + l)(t2 - Qt + 6) 581 C H A P T E R 6 SECTION 6.1 1. (a) T (b) T (c) F (d) F (e) F (f) F (g) F 2. (x,y) = 8 + 5i, x = V7, y = vT4, and x + y = >/&. ( h ) T e'-l l l + 3e^ 3- (/,o) = 1,| | / | | = ^ , 11011 = ^ ^ - ^ , and | | / + o| 16. (b) No SECTION 6.2 1. (a) F (b) T (c) T (d) F (e) T (f) F (g) T 2. For each part the orthonormal basis and the Fourier coefficients are given. (b) { f (1,1,1), f (-2,1,1), f (0, -1,1)}; 2 # , - f , f . ( c ) { l , 2 V 3 ( x - i ) , 6 V 5 ( x 2 - x + i ) } ; f , ^ , 0. (e) {(2,-1,-2,4), ^ ( - 4 , 2 , - 3 , 1 ) , ^ ( - 3 , 4 , 9 , 7 ) } ; 10, 3 ^ , vTo5 f i ( - ! O ' i f c O - 8 - i J l G = ) ' i M ~ ^ ( i ) { y f S i n t , ^ c o s t , ^ T ( l - l s i n t ) , v / ^ ' ( t + J c o s t - f ) } ; Ji(2n + 2), -4yjl, y/^(l + n), y ^ (k) { ^ ( - 4 , 3 - 2 i , i , l - 4 i ) , ^ ( 3 - i , - 5 i , - 2 + 4i,2 + i), - ^ ( - 1 7 - i, -9 + 8i, -18 + 16i, - 9 + 8i)}; y/4l(-l - i), V60i-1 + 2i), V/Il60(l + i) ( m ) { ^ I ( a - V i 1 -4% 11 9i - 5 - 1 1 8 i - 7 - 2 6 i ^39063 V ~uu _ 5 8 ; v/18(2 + i), v / 246( - l - i ) , 0 4. 5 - L =span({( i , - i ( l+ i ) , l )} ) 5. SQ is the plane through the origin that is perpendicular to xo; S1' is the line through the origin that is perpendicular to the plane containing xi and x2. 29 1 9- w h ( * ( b ) * I ' M- 7 5 582 Answers to Selected Exercises SECTION 6.3 1. (a) T (b) F (c) F (d) T (e) F (f) T (g) T 2. (a) y = (1, -2,4) (c) y = 210x2 - 204x + 33 3. (a) T* (x) = (11, -12) (c) r (/(*)) = 12 + 6t 14. T*(x) = (x,z)y 20. (a) The linear function is y = 2t + 5/2 with 7 = 1, and the quadratic function is y = t2 /3 - 4t/3 + 2 with 7 = 0. (b) The linear function is y = 1.25t + 0.55 with E = 0.3, and the quadratic function is t2/56 + 15t/14 + 239/280 with E = 0.22857 (approximation). 21. The spring constant is approximately 2.1. 22. (a) x = f , y = f, z (d) x = , y = i , z = , w = - ^ SECTION 6.4 1. (a) T (b) F (c) F (d) T (e) T (f) T (g) F 2. (a) T is self-adjoint. An orthonormal basis of eigenvectors is (h )T ( T=(1, 2), F(2, 1) , with corresponding eigenvalues 6 and 1. v5 V5 (c) T is normal, but not self-adjoint. An orthonormal basis of eigenvectors is < -(1 + i, V2), - (1 + i, V2) > with corresponding eigenvalues 2 + l + i a n d 2 _ i + ! . V2 V2 (e) T is self-adjoint. An orthonormal basis of eigenvectors is V2V1 o J ' ^ V o V ' v ^ V 1 0 / ' V 2 with corresponding eigenvalues 1, 1, 1, 1. SECTION 6.5 ( d ) T - 1 0 0 1 l . (a) T (g )F ( b ) F ( h ) F (c )F 0 ) F (e) F (f) T ' - K - i and D = 3 0N 0 - 1 ( d ) P = .1 73 1 V6 v^3 1 1 V6 v^ 3 V 0 -is j-J (-2 0 0N and D = I 0 - 2 0 0 0 4, 4. T2 is normal for all z G C, T2 is self-adjoint if and only if z G .ft, and T2 is unitary if and only if |.z| = 1. 5. Only the pair of matrices in (d) are unitarily equivalent. Answers to Selected Exercises 25. 2 0 - 0 ) 26. ( a ) t f - f ( b ) V + ! 27. ( a ) x = - ^ x ' + -^ 2 / ' and y = - L * ' - y' 583 The new quadratic form is 3(x') (y') . (c) x = T=X H7=2/ and y = -=x Hp=y V ' vT3 vT3 / l3 /l3 The new quadratic form is 5(x') 8(y') . 29. (c) Q = / l 1 _ 1 75 73 1 V3 6 6 and R = /v/2 V2 2v^ 0 /3 ^ V 0 0 ^ y 0 4- ^ , U 73 3 / (e) xi = 3, X2 = -5, X3 = 4 SECTION 6.6 1. (a) F (b) T (c) T (d) F (e) F 2. ForW = span({(l,2)}), Fh = ( ) 3. (2) (a) Ti(o,6) = | ( a + 6,o + 6) andT2(a,6) = ( a - o , - a + &) (d) Ti(a,6,c) = i ( 2 a - 6 - c , - a + 2 6 - c , - a - 6 + 2c) and T2(a, b,c) = | ( a + 6 + c,a + b + c,a + b + c) SECTION 6.7 1. (a) F (b) F (c) T (d) T (e) F (f) F (g) T 2-( a ) u i = G ) ' u 2 = ( i ) ' u i = ^ ( i j ' M 2 = ^ ( _ j ) ' u 3 = ^ 584 Answers to Selected Exercises (c) l v/lll I vTT) V ~4U 1 V'2 I V2 -2* o -i 0 0 1 73 _ u 2 vTo 2 yio 1 v2 y/2 (e) 2 1-t 2 ii 2 -I 4-i 2 >/ 0 0 0 2 ^6 l + i 4. (a) WP = 1 J_ / s/&+s/2 V2 75 / 2 1 1 J 1 -V+^2 75 N/2/ 2 1 -i 2 -N/S+V/2 2 2 5. (a) Tf(x,y,2) = x + U + z y - z (c) T^a + 6 sin a: + ccosx) = T' a 6sinx + ccosx) = a (26 + c) sin re + ( 6 + 2c) cos x 2 + ' ' I ' * S G I - i I - l o > ! ' -2 3 1 3 - 2 1 w S C r 1 1 + i i 7. (a) Zi = N(T)1 =R 2 and Z2 = R(T) = span{(l, 1,1),(0,1,-1)} (c) Zx = N(T)l = V and Z2 = R(T) - V 8. (a) No solution (e) T (f) F 2 VI SECTION 6.8 1. (a) F (b) K (c) T (d) F ( g ) F ( h ) F ( i )T ( j )F 4. (a) Yes (b) No (c) No (d) Yes 5. (a) 17. (a) and (b) 18. Same as Exercise 17(c) 22. (a) Q= (i '?! i.i f 0 0 l 0 0 0 0 0 0 0 0 1 0 0 1/ (c) (c) (e) Yes (f) No / 0 0 0 ON 1 0 - 4 0 0 0 0 0 - 2 0 - 8 0/ 0 (b) Q = 1 - * and D = /0 0 1 N (c) Q = I 0 1 -0.25 1 0 2 ; and -1 0 0 N 0 4 0 0 0 6.75, / 1 > V2 0 U J M 1 w V2 0 V~3j/ Answers to Selected Exercises SECTION 6.9 / 7. (P.)'1 = v T ^ V v T ^ 0 0 1 0 0 1 0 0 V v T s/T^v2/ SECTION 6.10 1. (a) F (b) F (c) T (d) F (e) F 2. (a) /l8 (c) approximately 2.34 4. (a) A ft* 84.74, ||A_1|| 17.01, and cond(A) 1441 (b) ||x - A~lb < ||A-11| Ax - 6|| a 0.17 and ^ - ^l 6' < c o n d ( A ) ' 6 - f ' U ^ l IA-1? 5. 0.001 < 'X *' < 10 585 6. ft - 2 = - , ||P|| = 2, and cond(P) = 2. SECTION 6.11 1. (a) F (b) T (c) T (d) F (e) T (g )F ( h ) F ( i )T ( j )F ( f )F 3- (b) {t ( y 1 : / C H 7. (c) There are six possibilities: (1) Any line through the origin if 0 = ip = 0 (2) < t I 0 I : t e ft > if 0 = 0 and tp = rr t G ft if 0 # 0 ( /cos ^ + F t I -sirn/-(4) i t I cos 0 - 1 t G ft > if0 = 7T and tp ^ ir t G ft > if^ = 7r and 0 ^ n (5) < t 1 : t G ft > if 0 = ^ = 7T 586 Answers to Selected Exercises { /sin 0(cos i> + 1)N t I sin0sint/> sint/>(cos0 + 1) t e ft otherwise C H A P T E R 7 SECTION 7.1 1. (a) T (b) F 2. (a) For A = 2, (c) For A = -1, (2 1 0N 3. (a) For A = 2, {2,-2x,x2} J = I 0 2 1 0 0 2, (f) F (g) T (h) T J = (c) For A = 1, SECTION 7.2 1. (a) T (b) T /Ax O O 2. J = I O A2 O 0 O A:i o o) n (o o) C 0 ) J = A 0 0 V' 1 0 1 0 0 1 0 0 0^ 0 1 (c )F ( d ) T ( e ) T ( f )F (g) F (2 1 0 0 0 0 0 2 1 0 0 0 0 0 2 0 0 0 0 0 0 2 1 0 0 0 0 0 2 0 0 0 0 0 0 2/ where A = A2 = /4 1 o ON 0 4 1 0 0 0 4 0 0 0 0 4/ and A3 = -3 0 0 -3 3. (a) - ( t - 2 ) 5 ( t - 3 ) ' (b) A, = 2 Ao = 3 ( h ) T (c) A2 = 3 (d) pi = 3 and p2 = 1 (e) (i) rank(Ui) = 3 and rank(U2) = 0 (ii) rank(U?) = 1 and rank(Ui) = 0 (iii) nullity(Ui) = 2 and nullity(U2) = 2 (iv) nullity(U?) = 4 and nullity(Ui) = 2 Answers to Selected Exercises 587 / I 0 ON 4. (a) J = I 0 2 1 0 0 2/ /o i o ON 0 0 0 0 and Q = (d) J = 5. (a) J = (c) J = ( d ) J = 0 = 0 0 2 0 0 0 0 2/ (1 1 0 ON 0 1 1 0 0 0 1 0 0 0 0 2/ /2 1 0 0 0 2 0 0 0 0 2 1 Q 0 0 2/ /2 1 0 ON 0 2 1 0 0 0 2 0 0 0 0 4/ 1 o W o 1 o o j ' l i o 1 l lN 2 1 2 1 - 1 o) (1 0 l - 1 1 - 2 i o 1 0 0 1 -IN l l 0/ and Q = and (3={2et,2tet,t2et,e2t} and /?= {6x,x3,2,x2} and ci + c2t) j 0 j + c2 1 (c! + c2t + c3r) 0 + (c2 + 2c3t) + 2c3 SECTION 7.3 1. (a) F (b) T (c) F (d) F (e) T (f) F (g) F (h) T (i) T 2. (a) it - l)(t - 3) (c) (t - l)2(t - 2) (d) it - 2)2 3. (a) t2 - 2 (c) (t - 2)2 (d) (t -l)it+ 1) 4. For (2), (a); for (3), (a) and (d) 5. The operators are To, I, and all operators having both 0 and 1 as eigenvalues. SECTION 7.4 1. (a) T (b) F 2. (a) (e) T (f) F (g) T (c) ( - l + n/3) 0 i-l-iV3) (e) /o 1 0 o - 2 0 0 0 0 0 0 1 ON 0 - 3 0/ 3. (a) t2 + 1 and t2 C = (c) t2 ~ t + 1 C -P = 1 0 0 0 /() - 1 0 0 1 0 0 0 0 0 0 0 Vo o o o/ (0 -1 0 0 1 1 0 0 0 0 0 - 1 0 0 1 1/ 0 ON (0 1 /0 0 - 1 o j ' v o o j ' l o - 1 (3= { l ,x ,-2x + x 2 , -3x + x3} 588 Index 589 I n d e x Absolute value of a complex num-ber, 558 Absorbing Markov chain, 304 Absorbing state, 304 Addition of matrices, 9 Addition of vectors, 6 Additive function, 78 Additive inverse of an element of a field, 553 of a vector, 12 Adjoint of a linear operator, 358-360 of a linear transformation, 367 of a matrix, 331, 359-360 uniqueness, 358 Algebraic multiplicity of an eigen-value, see Multiplicity of an eigenvalue Algebraically closed field, 482, 561 Alternating n-linear function, 239 Angle between two vectors, 202, 335 Annihilator of a subset, 126 of a vector, 524, 528 Approximation property of an or-thogonal projection, 399 Area of a parallelogram, 204 Associated quadratic form, 389 Augmented matrix, 161, 174 Auxiliary polynomial, 131, 134, 137-140 Axioms of the special theory of relativity, 453 Axis of rotation, 473 Back substitution, 186 Backward pass, 186 Basis, 43-49, 60-61, 192-194 cyclic, 526 dual, 120 Jordan canonical, 483 ordered, 79 orthonormal, 341, 346-347, 372 rational canonical, 526 standard basis for Fn, 43 standard basis for Pn(P), 43 standard ordered basis for F', 79 standard ordered basis for P(F), 79 uniqueness of size, 46 Bessel's inequality, 355 Bilinear form, 422-433 diagonalizable, 428 diagonalization, 428-435 index, 444 invariants, 444 matrix representation, 424-428 product with a scalar, 423 rank, 443 signature, 444 sum, 423 symmetric, 428-430, 433-435 vector space, 424 Cancellation law for vector addi-tion, 11 Cancellation laws for a field, 554 Canonical form Jordan, 483-516 rational, 526-548 for a symmetric matrix, 446 Cauchy-Schwarz inequality, 333 Cayley-Hamilton theorem for a linear operator, 317 for a matrix, 318, 377 Chain of sets, 59 Change of coordinate matrix, 112-115 Characteristic of a field, 23, 41, 42, 430, 449, 555 Characteristic polynomial, 373 590 Index 1 of a linear operator, 249 of a matrix, 248 Characteristic value, see Eigenvalue Characteristic vector, see Eigen-vector Classical adjoint of an n x n matrix, 231 of a 2 x 2 matrix, 208 Clique, 94, 98 Closed model of a simple econ-omy, 176-178 Closure under addition, 17 under scalar multiplication, 17 Codomain, 551 Coefficient matrix of a system of linear equations, 169 Coefficients Fourier, 119 of a differential equation, 128 of a linear combination, 24, 43 of a polynomial, 9 Cofactor, 210, 232 Cofactor expansion, 210, 215, 232 Column of a matrix, 8 Column operation, 148 Column sum of matrices, 295 Column vector, 8 Companion matrix, 526 Complex number, 556 561 absolute value, 558 conjugate, 557 fundamental theorem of alge-bra, 482, 560 imaginary part, 556 real part, 556 Composition of functions, 552 of linear transformations, 86 89 Condition number, 469 Conditioning of a system of linear equations, 464 Congruent matrices, 426, 445, 451 Conic sections, 388 392 Conjugate linear property, 333 Conjugate of a complex number, 557 Conjugate transpose of a matrix, 331, 359 360 Consistent system of linear equa-tions, 169 Consumption matrix, 177 Convergence of matrices, 284 288 Coordinate function, 119 120 Coordinate system left-handed, 203 right-handed, 202 Coordinate vector, 80, 91, 110-111 Corresponding homogeneous sys-tem of linear equations, 172 Coset, 23, 109 Cramer's rule, 224 Critical point, 439 Cullen, Charles G., 470 Cycle of generalized eigenvectors, 488-491 end vector, 488 initial vector, 488 length, 488 Cyclic basis, 526 Cyclic subspace, 313-317 Degree of a polynomial, 10 Determinant, 199-243 area of a parallelogram, 204 characterization of, 242 cofactor expansion, 210, 215, 232 Cramer's rule, 224 of an identity matrix, 212 of an invertible matrix, 223 of a linear operator, 258, 474, 476-477 of a matrix transpose, 224 of a n n x n matrix, 210, 232 n-dimensional volume, 226 properties of, 234 236 of a square matrix, 367, 394 of a 2 x 2 matrix, 200 uniqueness of, 242 of an upper triangular matrix, 218 Index 591 volume of a parallelepiped, 226 Wronskian, 232 Diagonal entries of a matrix, 8 Diagonal matrix, 18, 97 Diagonalizable bilinear form, 428 Diagonalizable linear operator, 245 Diagonalizable matrix, 246 Diagonalization of a bilinear form, 428-435 problem, 245 simultaneous, 282, 325, 327, 376, 405 of a symmetric matrix, 431-433 test, 269, 496 Diagonalize, 247 Differentiable function, 129 Differential equation, 128 auxiliary polynomial, 131, 134, 137-140 coefficients, 128 homogeneous, 128,137-140, 523 linear, 128 nonhomogeneous, 142 order, 129 solution, 129 solution space, 132, 137-140 system, 273, 516 Differential operator, 131 null space, 134-137 order, 131, 135 Dimension, 47-48, 50-51,103,119, 425 Dimension theorem, 70 Direct sum of matrices, 320-321, 496, 545 of subspaces, 22, 58, 98, 275-279, 318, 355, 366, 394, 398, 401, 475-478, 494, 545 Disjoint sets, 550 Distance, 340 Division algorithm for polynomi-als, 562 Domain, 551 Dominance relation, 95-96, 99 Dot diagram of a Jordan canonical form, 498-500 of a rational canonical form, 535 -539 Double dual, 120, 123 Dual basis, 120 Dual space, 119-123 Economics, see Leontief, Wassily Eigenspace generalized, 485-491 of a linear operator or matrix, 264 Eigenvalue of a generalized eigenvector, 484 of a linear operator or matrix, 246, 371-374, 467-470 multiplicity, 263 Eigenvector generalized, 484-491 of a linear operator or matrix, 246, 371-374 Einstein, Albert, see Special the-ory of relativity Element, 549 Elementary column operation, 148, 153 Elementary divisor of a linear operator, 539 of a matrix, 541 Elementary matrix, 149-150, 159 Elementary operation, 148 Elementary row operation, 148, 153, 217 Ellipse, see Conic sections Empty set, 549 End vector of a cycle of general-ized eigenvectors, 488 Entry of a matrix, 8 Equality of functions, 9, 551 of matrices, 9 of n-tuples, 8 of polynomials, 10 of sets, 549 Equilibrium condition for a sim-ple economy, 177 Equivalence relation, 107, 551 congruence, 449, 451 592 Index unitary equivalence, 394, 472 Equivalent systems of linear equa-tions, 182-183 Euclidean norm of a matrix, 467-470 Euler's formula, 132 Even function, 15, 21, 355 Exponential function, 133 140 Exponential of a matrix, 312, 515 Extremum, see Local extrcmum Field, 553 555 algebraically closed, 482, 561 cancellation laws, 554 characteristic, 23, 41, 42, 430, 449, 555 of complex numbers, 556 561 product of elements, 553 of real numbers, 549 sum of elements, 553 Field of scalars, 6-7, 47 Finite-dimensional vector space, 46-51 Fixed probability vector, 301 Forward pass, 186 Fourier, Jean Baptiste, 348 Fourier coefficients, 119, 348, 400 Frobenius inner product, 332 Function, 551-552 additive, 78 alternating n-linear, 239 codomain of, 551 composite, 552 coordinate function, 119-120 differentiablc, 129 domain of, 551 equality of, 9, 551 even, 15, 21, 355 exponential, 133-140 image of, 551 imaginary part of, 129 inverse, 552 invertible, 552 linear, sec Linear transforma-tion ?>.-linear, 238 242 norm, 339 odd, 21, 355 one-to-one, 551 onto, 551 polynomial, 10, 51-53, 569 preimage of, 551 range of, 551 real part of, 129 restriction of, 552 sum of, 9 vector space, 9 Fundamental theorem of algebra, 482, 560 Gaussian elimination, 186 187 back substitution, 186 backward pass, 186 forward pass, 186 General solution of a system of linear equations, 189 Generalized eigenspace, 485-491 Generalized eigenvector, 484-491 Generates, 30 Generator of a cyclic subspace, 313 Geometry, 385, 392, 436, 472 478 Gerschgorin's disk theorem, 296 Gram Schmidt process, 344, 396 Gramian matrix, 376 Hardy Weinberg law, 307 Hermitian operator or matrix, see Self-adjoint linear operator or matrix Hessian matrix, 440 Homogeneous linear differential equa-tion, 128, 137 140, 523 Homogeneous polynomial of de-gree two, 433 Homogeneous system of linear equa-tions, 171 Hooke's law, 128, 368 Householder operator, 397 Identity element in C, 557 in a field, 553, 554 Identity matrix, 89, 93, 212 Identity transformation, 67 Ill-conditioned system, 464 Index 593 Image, see Range Image of an element, 551 Imaginary number, 556 Imaginary part of a complex number, 556 of a function, 129 Incidence matrix, 94-96, 98 Inconsistent system of linear equa-tions, 169 Index of a bilinear form, 444 of a matrix, 445 Infinite-dimensional vector space, 47 Initial probability vector, 292 Initial vector of a cycle of gener-alized eigenvectors, 488 Inner product, 329-336 Frobenius, 332 on H, 335 standard, 330 Inner product space complex, 332 H, 332, 343, 348-349, 380, 399 real, 332 Input-output matrix, 177 Intersection of sets, 550 Invariant subspace, 77-78. 313 315 Invariants of a bilinear form, 444 of a matrix, 445 Inverse of a function, 552 of a linear transformation, 99 102, 164-165 of a matrix, 100 102, 107, 161-164 Invertible function, 552 Invertible linear transformation, 99 102 Invertible matrix, 100 102, 111, 223, 469 Irreducible polynomial, 525, 567 569 Isometry, 379 Isomorphic vector spaces, 102 105 Isomorphism, 102-105, 123, 425 Jordan block, 483 Jordan canonical basis, 483 Jordan canonical form dot diagram, 498 500 of a linear operator, 483-516 of a matrix, 491 uniqueness, 500 Kernel, sec Null space Kronecker delta, 89, 335 Lagrange interpolation formula, 51 53, 125, 402 Lagrange polynomials, 51, 109, 125 Least squares approximation, 360 364 Least squares line, 361 Left shift, operator, 76 Left-handed coordinate system, 203 Left-multiplication transformation, 92-94 Legendre polynomials, 346 Length of a cycle of generalized eigenvectors, 488 Length of a vector, see Norm Leontief closed model, 176-178 open model, 178 179 Leontief, Wassily, 176 Light second, 452 Limit of a sequence of matrices, 284 288 Linear combination, 24-26, 28-30, 39 uniqueness of coefficients, 43 Linear dependence, 36 40 Linear differential equation, 128 Linear equations, see System of linear equations Linear functional, 119 Linear independence, 37 40, 59 61, 342 Linear operator, (see also Linear transformation), 112 adjoint, 358 360 characteristic polynomial, 249 594 Index determinant, 258, 474, 476 477 diagonalizable, 245 diagonalize, 247 differential, 131 differentiation, 131, 134-137 eigenspace, 264, 401 eigenvalue, 246, 371 -374 eigenvector, 246, 371-374 elementary divisor, 539 Householder operator, 397 invariant subspace, 77 78, 313-315 isometry, 379 Jordan canonical form, 483-516 left shift, 76 Lorentz transformation, 454-461 minimal polynomial, 516-521 nilpotent, 512 normal, 370, 401-403 orthogonal, 379-385, 472-478 partial isometry, 394, 405 positive definite, 377-378 positive semidefinite, 377 378 projection, 398 403 projection on a subspace, 86, 117 projection on the x-axis, 66 quotient space, 325-326 rational canonical form, 526 548 reflection, 66, 113, 117, 387, 472-478 right shift, 76 rotation, 66, 382, 387, 472-478 self-adjoint, 373. 401-403 simultaneous diagonalization, 282, 405 spectral decomposition, 402 spectrum, 402 unitary, 379-385, 403 Linear space, see Vector space Linear transformation, (see also Linear operator), 65 adjoint, 367 composition, 86-89 identity, 67 image, see Range inverse, 99-102, 164-165 invertible, 99 102 isomorphism, 102-105, 123, 425 kernel, see Null space left-multiplication, 92-94 linear functional, 119 matrix representation, 80, 88 92, 347, 359 null space, 67 69, 134-137 nullity, 69-71 one-to-one, 71 onto, 71 product with a scalar, 82 pseudoinverse, 413 range, 67-69 rank, 69-71, 159 restriction, 77 -78 singular value, 407 singular value theorem, 406 sum, 82 transpose, 121, 126, 127 vector space of, 82, 103 zero, 67 Local extremum, 439, 450 Local maximum, 439, 450 Local minimum, 439, 450 Lorentz transformation, 454-461 Lower triangular matrix, 229 Markov chain, 291, 304 Markov process, 291 Matrix, 8 addition, 9 adjoint, 331, 359-360 augmented, 161, 174 change of coordinate, 112-115 characteristic polynomial, 248 classical adjoint, 208, 231 coefficient, 169 cofactor, 210, 232 column of, 8 column sum, 295 companion, 526 condition number, 469 congruent, 426, 445, 451 conjugate transpose, 331, 359 360 consumption, 177 Index 595 convergence, 284-288 determinant of, 200, 210, 232, 367, 394 diagonal, 18, 97 diagonal entries of, 8 diagonalizable, 246 diagonalize, 247 direct sum, 320-321, 496, 545 eigenspace, 264 eigenvalue, 246, 467-470 eigenvector, 246 elementary, 149-150, 159 elementary divisor, 541 elementary operations, 148 entry, 8 equality of, 9 Euclidean norm, 467-470 exponential of, 312, 515 Gramian, 376 Hessian, 440 identity, 89 incidence, 94-96, 98 index, 445 input-output, 177 invariants, 445 inverse, 100-102, 107, 161-164 invertible, 100-102, 111, 223, 469 Jordan block, 483 Jordan canonical form, 491 limit of, 284-288 lower triangular, 229 minimal polynomial, 517-521 multiplication with a scalar, 9 nilpotent, 229, 512 norm, 339, 467-470, 515 normal, 370 orthogonal, 229, 382-385 orthogonally equivalent, 384-385 permanent of a 2 x 2, 448 polar decomposition, 411-413 positive definite, 377 positive semidefinite, 377 product, 87-94 product with a scalar, 9 pseudoinverse, 414 rank, 152-159 rational canonical form, 541 reduced row echelon form, 185, 190-191 regular, 294 representation of a bilinear form, 424-428 representation of a linear trans-formation, 80, 88-92, 347, 359 row of, 8 row sum, 295 scalar, 258 self-adjoint, 373, 467 signature, 445 similarity, 115, 118, 259, 508 simultaneous diagonalization, 282 singular value, 410 singular value decomposition, 410 skew-symmetric, 23, 229, 371 square, 9 stochastic, see Transition ma-trix submatrix, 230 sum, 9 symmetric, 17, 373, 384, 389, 446 trace, 18, 20, 97, 118, 259, 281, 331, 393 transition, 288-291, 515 transpose, 17, 20, 67, 88, 127, 224, 259 transpose of a matrix inverse, 107 transpose of a product, 88 unitary, 229, 382-385 unitary equivalence, 384-385, 394, 472 upper triangular, 21, 218, 258, 370, 385, 397 Vandermonde, 230 vector space, 9, 331, 425 zero, 8 Maximal element of a family of sets, 58 Maximal linearly independent sub-set, 59-61 Maximal principle, 59 596 Index Member, see Element Michelson-Morley experiment, 451 Minimal polynomial of a linear operator, 516 521 of a matrix, 517-521 uniqueness, 516 Minimal solution to a system of linear equations, 364-365 Monic polynomial, 567 569 Multiplicative inverse of an ele-ment of a field, 553 Multiplicity of an eigenvalue, 263 Multiplicity of an elementary di-visor, 539, 541 n-dimensional volume, 226 n-linear function, 238-242 n-tuple, 7 equality, 8 scalar multiplication, 8 sum, 8 vector space, 8 Nilpotent linear operator, 512 Nilpotent matrix, 229, 512 Nonhomogeneous linear differen-tial equation, 142 Nonhomogeneous system of linear equations, 171 Nonnegative vector, 177 Norm Euclidean, 467 470 of a function, 339 of a matrix, 339, 467-470, 515 of a vector, 333-336, 339 Normal equations, 368 Normal linear operator or matrix, 370, 401-403 Normalizing a vector, 335 Null space, 67 69, 134-137 Nullity, 69-71 Numerical methods conditioning, 464 QR factorization, 396 397 Odd function, 21, 355 One-to-one function, 551 One-to-one linear transformation, 71 Onto function, 551 Onto linear transformation, 71 Open model of a simple economy, 178-179 Order of a differential equation, 129 of a differential operator, 131, 135 Ordered basis, 79 Orientation of an ordered basis, 202 Orthogonal complement, 349, 352, 398 401 Orthogonal equivalence of matrices, 384-385 Orthogonal matrix, 229, 382 385 Orthogonal operator, 379-385, 472 478 on R2, 387-388 Orthogonal projection, 398-403 Orthogonal projection of a vector, 351 Orthogonal subset, 335, 342 Orthogonal vectors, 335 Orthonormal basis, 341, 346 347, 372 Orthonormal subset, 335 Parallel vectors, 3 Parallelogram area of, 204 law, 2, 337 Parseval's identity, 355 Partial isometry, 394, 405 Pendular motion, 143 Penrose conditions, 421 Periodic motion of a spring, 127, 144 Permanent of a 2 x 2 matrix, 448 Perpendicular vectors, see Orthog-onal vectors Physics Hooke's law, 128, 368 pendular motion, 143 periodic motion of a spring, 144 special theory of relativity, 451-461 n . T-Index 597 spring constant, 368 Polar decomposition of a matrix, 411-413 Polar identities, 338 Polynomial, 9 annihilator of a vector, 524, 528 auxiliary, 131, 134, 137-140 characteristic, 373 coefficients of, 9 degree of a, 10 division algorithm, 562 equality, 10 function, 10, 51-53, 569 fundamental theorem of alge-bra, 482, 560 homogeneous of degree two, 433 irreducible, 525, 567-569 Lagrange, 51, 109, 125 Legendre, 346 minimal, 516-521 monic, 567-569 product with a scalar, 10 quotient, 563 relatively prime, 564 remainder, 563 splits, 262, 370, 373 sum, 10 trigonometric, 399 unique factorization theorem, 568 vector space, 10 zero, 9 zero of a, 62, 134, 560, 564 Positive definite matrix, 377 Positive definite operator, 377 378 Positive semidefinite matrix, 377 Positive semidefinite operator, 377-378 Positive vector, 177 Power set, 59 Preimage of an element, 551 Primary decomposition theorem, 545 Principal axis theorem, 390 Probability, see Markov chain Probability vector, 289 fixed, 301 initial, 292 Product of a bilinear form and a scalar, 423 of complex numbers, 556 of elements of a field, 553 of a linear transformation and scalar, 82 of matrices, 87-94 of a matrix and a scalar, 9 of a vector and a scalar, 7 Projection on a subspace, 76, 86, 98, 117, 398-403 on the x-axis, 66 orthogonal, 398-403 Proper subset, 549 Proper value, see Eigenvalue Proper vector, see Eigenvector Pseudoinverse of a linear transformation, 413 of a matrix, 414 Pythagorean theorem, 337 QR factorization, 396-397 Quadratic form, 389, 433-439 Quotient of polynomials, 563 Quotient space, 23, 58, 79, 109, 325-326 Range, 67-69, 551 Rank of a bilinear form, 443 of a linear transformation, 69-71, 159 of a matrix, 152-159 Rational canonical basis, 526 Rational canonical form dot diagram, 535-539 elementary divisor, 539, 541 of a linear operator, 526-548 of a matrix, 541 uniqueness, 539 Rayleigh quotient, 467 Real part of a complex number, 556 of a function, 129 Reduced row echelon form of a matrix, 185, 190-191 598 Index Reflection, 66, 117, 472-478 of R2, 113, 382-383, 387, 388 Regular transition matrix, 294 Relation on a set, 551 Relative change in a vector, 465 Relatively prime polynomials, 564 Remainder, 563 Replacement theorem, 45-46 Representation of a linear trans-formation by a matrix, 80 Resolution of the identity opera-tor, 402 Restriction of a function, 552 of a linear operator on a sub-space, 77 78 Right shift operator, 76 Right-handed coordinate system, 202 Rigid motion, 385 387 in the plane, 388 Rotation, 66, 382, 387, 472-478 Row of a matrix, 8 Row operation, 148 Row sum of matrices, 295 Row vector, 8 Rudin. Walter, 560 Saddle point, 440 Scalar, 7 Scalar matrix, 258 Scalar multiplication, 6 Schur's theorem for a linear operator, 370 for a matrix, 385 Second derivative test, 439 443, 450 Self-adjoint linear operator or ma-trix, 373, 401-403, 467 Sequence, 11 Set, 549 551 chain, 59 disjoint, 550 element of a, 549 empty, 549 equality of, 549 equivalence relation, 107, 394, 449, 451 equivalence relation on a, 551 intersection, 550 linearly dependent, 36-40 linearly independent, 37-40 orthogonal, 335, 342 orthonormal, 335 power, 59 proper subset, 549 relation on a, 551 subset, 549 union, 549 Signature of a bilinear form, 444 of a matrix, 445 Similar matrices, 115, 118, 259, 508 Simpson's rule, 126 Simultaneous diagonalization, 282, 325, 327, 376, 405 Singular value of a linear transformation, 407 of a matrix, 410 Singular value decomposition of a matrix, 410 Singular value decomposition the-orem for matrices, 410 Singular value theorem for linear transformations, 406 Skew-symmetric matrix, 23, 229, 371 Solution of a differential equation, 129 minimal, 364-365 to a system of linear equations, 169 Solution set of a system of linear equations, 169, 182 Solution space of a homogeneous differential equation, 132, 137 140 Space time coordinates, 453 Span, 30, 34, 343 Special theory of relativity, 451-461 axioms, 453 Lorentz transformation, 454 461 spacetime coordinates, 453 Index 599 time contraction, 459-461 Spectral decomposition, 402 Spectral theorem, 401 Spectrum, 402 Splits, 262, 370, 373 Spring, periodic motion of, 127, 144 Spring constant, 368 Square matrix, 9 Square root of a unitary operator, 393 Standard basis for Fn, 43 for P(F), 43 Standard inner product on Fn, 330 Standard ordered basis for Fn, 79 for Pn(F), 79 Standard representation of a vec-tor space, 104-105 States absorbing, 304 of a transition matrix, 288 Stationary vector, see Fixed prob-ability vector Statistics, see Least squares ap-proximation Stochastic matrix, see Transition matrix Stochastic process, 291 Submatrix, 230 Subset, 549 linearly dependent, 36-40 linearly independent, 59-61 maximal linearly independent, 59-61 orthogonal, 335, 342 orthogonal complement of a, 349, 352, 398- 401 orthonormal, 335 span of a, 30, 34, 343 sum, 22 Subspace, 16-19, 50-51 cyclic, 313-317 dimension of a, 50-51 direct sum, 22, 58, 98, 275-279, 318, 355, 366, 394,398,401, 475-478, 494, 545 generated by a set, 30 invariant, 77-78 sum, 275 zero, 16 Sum of bilinear forms, 423 of complex numbers, 556 of elements of a field, 553 of functions, 9 of linear transformations, 82 of matrices, 9 of n-tuples, 8 of polynomials, 10 of subsets, 22 of vectors, 7 Sum of subspaces, (see also Direct sum, of subspaces), 275 Sylvester's law of inertia for a bilinear form, 443 for a matrix, 445 Symmetric bilinear form, 428-430, 433-435 Symmetric matrix, 17, 373, 384, 389, 446 System of differential equations, 273, 516 System of linear equations, 25-30, 169 augmented matrix, 174 coefficient matrix, 169 consistent, 169 corresponding homogeneous sys-tem, 172 equivalent, 182-183 Gaussian elimination, 186187 general solution, 189 homogeneous, 171 ill-conditioned, 464 inconsistent, 169 minimal solution, 364-365 nonhomogeneous, 171 solution to, 169 well-conditioned, 464 T-annihilator, 524, 528 T-cyclic basis, 526 ' 7 600 T-cyclic subspace, 313-317 T-invariant subspace, 77 78, 313-315 Taylor's theorem, 441 Test for diagonalizability, 496 Time contraction, 459-461 Trace of a matrix, 18, 20, 97, 118, 259, 281, 331, 393 Transition matrix, 288-291, 515 regular, 294 states, 288 Translation, 386 Transpose of an invertible matrix, 107 of a linear transformation, 121, 126, 127 of a matrix, 17, 20, 67, 88, 127, 224, 259 Trapezoidal rule, 126 Triangle inequality, 333 Trigonometric polynomial, 399 Trivial representation of zero vec-tor, 36-38 Union of sets, 549 Unique factorization theorem for polynomials, 568 Uniqueness of adjoint, 358 of coefficients of a linear com-bination, 43 of Jordan canonical form, 500 of minimal polynomial, 516 of rational canonical form, 539 of size of a basis, 46 Unit vector, 335 Unitary equivalence of matrices, 384-385, 394, 472 Unitary matrix, 229, 382 385 Unitary operator, 379-385, 403 Upper triangular matrix, 21, 218, 258, 370, 385, 397 Vandcrmonde matrix, 230 Vector, 7 additive inverse of a, 12 annihilator of a, 524, 528 column, 8 Index coordinate, 80, 91, 110-111 fixed probability, 301 Fourier coefficients, 119, 348, 400 initial probability, 292 linear combination, 24 nonnegative, 177 norm, 333-336, 339 normalizing, 335 orthogonal, 335 orthogonal projection of a, 351 parallel, 3 perpendicular, see Orthogonal vectors positive, 177 probability, 289 product with a scalar, 8 Rayleigh quotient, 467 row, 8 sum, 7 unit, 335 zero, 12, 36-38 Vector space, 6 addition, 6 basis, 43-49, 192-194 of bilinear forms, 424 of continuous functions, 18, 67, 119, 331, 345, 356 of cosets, 23 dimension, 47-48, 103, 119, 425 dual, 119 123 finite-dimensional, 46-51 of functions from a set into a field, 9, 109, 127 infinite-dimensional, 47 of infinitely differentiable func-tions, 130 137, 247, 523 isomorphism, 102-105, 123, 425 of linear transformations, 82, 103 of matrices, 9, 103, 331, 425 of n-tuples, 8 of polynomials, 10, 86, 109 quotient, 23, 58, 79, 109 scalar multiplication, 6 of sequences, 11, 109, 356, 369 subspace, 16 19, 50-51 zero, 15 zero vector of a, 12 = Index Volume of a parallelepiped, 226 Wade, William R., 439 Well-conditioned system, 464 Wilkinson, J. H., 397 Wronskian, 232 Z2, 16, 42, 429, 553 Zero matrix, 8 601 Zero of a polynomial, 62, 134, 560, 564 Zero polynomial, 9 Zero subspace, 16 Zero transformation, 67 Zero vector, 12, 36-38 trivial representation, 36- 38 Zero vector space, 15 L i s t o f S y m b o l s ( c o n t i n u e d ) Si+S2 span (5) m z T'1 Tt T* To T* Te Tw tr(A) V* v / w W, + + Wfc E t i W , Wj w 2 w, w* INI [xb {x,y) z2 z 0 the sum of sets S and S2 page 22 the span of the set S page 30 the orthogonal complement of the set S page 349 the matrix representation of T in basis (3 page 80 the matrix representation of T in bases (3 and 7 page 80 the inverse of the linear transformation T page 99 the pseudoinverse of the linear transformation T page 413 the adjoint of the linear operator T page 358 the zero transformation page 67 the transpose of the linear transformation T page 121 the rotation transformation by 6 page 66 the restriction of T to a subspace W page 314 the trace of the matrix A page 18 the dual space of the vector space V page 119 the quotient space of V modulo W page 23 the sum of subspaces W] through Wfc page 275 the sum of subspaces Wi through Wfc page 275 the direct sum of subspaces Wi and W2 page 22 the direct sum of subspaces Wi through Wfc page 275 the norm of the vector x page 333 the coordinate vector of x relative to 0 page 80 the inner product of x and y page 330 the field consisting of 0 and 1 page 553 the complex conjugate of z page 557 the zero vector page 7 Pearson Education Prentice Hall Upper Saddle River, NJ 07458 www.prenhall.com 1-3031-100101102-200201-300301-350351-400401-450451-500500-end