The Theory of Matrices, Second Edition: With Applications (Computer Science and Scientific Computing)

Cover page......Page 1
Title page......Page 5
Contents......Page 9
Preface......Page 15
1 Matrix Algebra......Page 19
1.1 Special Types of Matrices ......Page 20
1.2 The Operations of Addition and Scalar Multiplication ......Page 22
1.3 Matrix Multiplication ......Page 25
1.4 Special Kinds of Matrices Related to Multiplication ......Page 28
1.5 Transpose and Conjugate Transpose ......Page 31
1.6 Submatrices and Partitions of a Matrix ......Page 34
1.7 Polynomials in a Matrix ......Page 37
1.8 Miscellaneous Exercises ......Page 39
2.1 Definition of the Determinant ......Page 41
2.2 Properties of Determinants ......Page 45
2.3 Cofactor Expansions ......Page 50
2.4 Laplace\'s Theorem ......Page 54
2.5 The Binet-Cauchy Formula ......Page 57
2.6 Adjoint and Inverse Matrices ......Page 60
2.7 Elementary Operations on Matrices ......Page 65
2.8 Rank of a Matrix ......Page 71
2.9 Systems of Linear Equations and Matrices ......Page 74
2.10 The LU Decomposition ......Page 79
2.11 Miscellaneous Exercises ......Page 81
3.1 Definition of a Linear Space ......Page 89
3.2 Subspaces ......Page 93
3.3 Linear Combinations ......Page 96
3.4 Linear Dependence and Independence ......Page 98
3.5 The Notion of a Basis ......Page 101
3.6 Sum and Direct Sum of Subspaces ......Page 105
3.7 Matrix Representation and Rank ......Page 109
3.8 Some Properties of Matrices Related to Rank ......Page 113
3.9 Change of Basis and Transition Matrices ......Page 116
3.10 Solution of Equations ......Page 118
3.11 Unitary and Euclidean Spaces ......Page 122
3.12 Orthogonal Systems ......Page 125
3.13 Orthogonal Subspaces ......Page 129
3.14 Miscellaneous Exercises ......Page 131
4 Linear Transformations and Matrices......Page 134
4.1 Linear Transformations ......Page 135
4.2 Matrix Representation of Linear Transformations ......Page 140
4.3 Matrix Representations, Equivalence, and Similarity ......Page 145
4.4 Some Properties of Similar Matrices ......Page 149
4.5 Image and Kernel of a Linear Transformation ......Page 151
4.6 Invertible Transformations ......Page 156
4.7 Restrictions, Invariant Subspaces, and Direct Sums of Transformations ......Page 160
4.8 Direct Sums and Matrices ......Page 163
4.9 Eigenvalues and Eigenvectors of a Transformation ......Page 165
4.10 Eigenvalues and Eigenvectors of a Matrix ......Page 170
4.11 The Characteristic Polynomial ......Page 173
4.12 The Multiplicities of an Eigenvalue ......Page 177
4.13 First Applications to Differential Equations ......Page 179
4.14 Miscellaneous Exercises ......Page 182
5.1 Adjoint Transformations ......Page 186
5.2 Normal Transformations and Matrices ......Page 192
5.3 Hermitian, Skew-Hermitian, and Definite Matrices ......Page 196
5.4 Square Root of a Definite Matrix and Singular Values ......Page 198
5.5 Congruence and the Inertia of a Matrix ......Page 202
5.6 Unitary Matrices ......Page 206
5.7 Polar and Singular-Value Decompositions ......Page 208
5.8 Idempotent Matrices (Projectors) ......Page 212
5.9 Matrices over the Field of Real Numbers ......Page 218
5.10 Bilinear, Quadratic, and Hermitian Forms ......Page 220
5.11 Finding the Canonical Forms ......Page 223
5.12 The Theory of Small Oscillations ......Page 226
5.13 Admissible Pairs of Matrices ......Page 230
5.14 Miscellaneous Exercises ......Page 235
6 The Jordan Canonical Form: A Geometric Approach......Page 238
6.1 Annihilating Polynomials ......Page 239
6.2 Minimal Polynomials ......Page 242
6.3 Generalized Eigenspaces ......Page 247
6.4 The Structure of Generalized Eigenspaces ......Page 250
6.5 The Jordan Theorem ......Page 254
6.6 Parameters of a Jordan Matrix ......Page 257
6.7 The Real Jordan Form ......Page 260
6.8 Miscellaneous Exercises ......Page 262
7.1 The Notion of a Matrix Polynomial ......Page 264
7.2 Division of Matrix Polynomials ......Page 266
7.3 Elementary Operations and Equivalence ......Page 271
7.4 A Canonical Form for a Matrix Polynomial ......Page 274
7.5 Invariant Polynomials and the Smith Canonical Form ......Page 277
7.6 Similarity and the First Normal Form ......Page 280
7.7 Elementary Divisors ......Page 283
7.8 The Second Normal Form and the Jordan Normal Form ......Page 287
7.9 The Characteristic and Minimal Polynomials ......Page 289
7.10 The Smith Form: Differential and Difference Equations ......Page 292
7.11 Miscellaneous Exercises ......Page 296
8 The Variational Method......Page 300
8.1 Field of Values. Extremal Eigenvalues of a Hermitian Matrix ......Page 301
8.2 Courant-Fischer Theory and the Rayleigh Quotient ......Page 304
8.3 The Stationary Property of the Rayleigh Quotient ......Page 307
8.4 Problems with Constraints ......Page 308
8.5 The Rayleigh Theorem and Definite Matrices ......Page 312
8.6 The Jacobi-Gundelfinger-Frobenius Method ......Page 314
8.7 An Application of the Courant-Fischer Theory ......Page 318
8.8 Applications to the Theory of Small Vibrations ......Page 320
9 Functions of Matrices......Page 322
9.1 Functions Defined on the Spectrum of a Matrix ......Page 323
9.2 Interpolatory Polynomials ......Page 324
9.3 Definition of a Function of a Matrix ......Page 326
9.4 Properties of Functions of Matrices ......Page 328
9.5 Spectral Resolution of f(A) ......Page 332
9.6 Component Matrices and Invariant Subspaces ......Page 338
9.7 Further Properties of Functions of Matrices ......Page 340
9 8 Sequences and Series of Matrices ......Page 343
9.9 The Resolvent and the Cauchy Theorem for Matrices ......Page 347
9.10 Applications to Differential Equations ......Page 352
9.11 Observable and Controllable Systems ......Page 358
9.12 Miscellaneous Exercises ......Page 363
10.1 The Notion of a Norm ......Page 368
10.2 A Vector Norm as a Metric: Convergence ......Page 372
10.3 Matrix Norms ......Page 376
10.4 Induced Matrix Norms ......Page 380
10.5 Absolute Vector Norms and Lower Bounds of a Matrix ......Page 385
10.6 The Gerigorin Theorem ......Page 389
10.7 Gerlgorin Disks and Irreducible Matrices ......Page 392
10.8 The Schur Theorem ......Page 395
10.9 Miscellaneous Exercises ......Page 398
11.1 Perturbations in the Solution of Linear Equations ......Page 401
11.2 Perturbations of the Eigenvalues of a Simple Matrix ......Page 405
11.3 Analytic Perturbations ......Page 409
11.4 Perturbation of the Component Matrices ......Page 411
11.5 Perturbation of an Unrepeated Eigenvalue ......Page 413
11.6 Evaluation of the Perturbation Coefficients ......Page 415
11.7 Perturbation of a Multiple Eigenvalue ......Page 417
12.1 The Notion of a Kronecker Product ......Page 424
12.2 Eigenvalues of Kronecker Products and Composite Matrices ......Page 429
12.3 Applications of the Kronecker Product to Matrix Equations ......Page 431
12.4 Commuting Matrices ......Page 434
12.5 Solutions of AX + XB = C ......Page 439
12.6 One-Sided Inverses ......Page 442
12.7 Generalized Inverses ......Page 446
12.8 The Moore-Penrose Inverse ......Page 450
12.9 The Best Approximate Solution of the Equation Ax = b ......Page 453
12.10 Miscellaneous Exercises ......Page 456
13.1 The Lyapunov Stability Theory and Its Extensions ......Page 459
13.2 Stability with Respect to the Unit Circle ......Page 469
13.3 The Bezoutian and the Resultant ......Page 472
13.4 The Hermite and the Routh-Hurwitz Theorems ......Page 479
13.5 The Schur-Cohn Theorem ......Page 484
13.6 Perturbations of a Real Polynomial ......Page 486
13.7 The Liénard-Chipart Criterion ......Page 488
13.8 The Markov Criterion ......Page 492
13.9 A Determinantal Version of the Routh-Hurwitz Theorem ......Page 496
13.10 The Cauchy Index and Its Applications ......Page 500
14 Matrix Polynomials......Page 507
14.1 Linearization of a Matrix Polynomial ......Page 508
14.2 Standard Triples and Pairs ......Page 511
14.3 The Structure of Jordan Triples ......Page 518
14.4 Applications to Differential Equations ......Page 524
14.5 General Solutions of Differential Equations ......Page 527
14.6 Difference Equations ......Page 530
14.7 A Representation Theorem ......Page 534
14.8 Multiples and Divisors ......Page 536
14.9 Solvents of Monic Matrix Polynomials ......Page 538
15 Nonnegative Matrices......Page 545
15.1 Irreducible Matrices ......Page 546
15.2 Nonnegative Matrices and Nonnegative Inverses ......Page 548
15.3 The Perron-Frobenius Theorem (I) ......Page 550
15.4 The Perron-Frobenius Theorem (II) ......Page 556
15.5 Reducible Matrices ......Page 561
15.6 Primitive and Imprimitive Matrices ......Page 562
15.7 Stochastic Matrices ......Page 565
15.8 Markov Chains ......Page 568
Appendix 1: A Survey of Scalar Polynomials ......Page 571
Appendix 2: Some Theorems and Notions from Analysis ......Page 575
Appendix 3: Suggestions for Further Reading ......Page 578
Index ......Page 581