The theory of matrices: with applications

Contents......Page 6
Preface......Page 12
1 Matrix Algebra......Page 16
1.1 Special Types of Matrices ......Page 17
1.2 The Operations of Addition and Scalar Multiplication ......Page 19
1.3 Matrix Multiplication ......Page 22
1.4 Special Kinds of Matrices Related to Multiplication ......Page 25
1.5 Transpose and Conjugate Transpose ......Page 28
1.6 Submatrices and Partitions of a Matrix ......Page 31
1.7 Polynomials in a Matrix ......Page 34
1.8 Miscellaneous Exercises ......Page 36
2.1 Definition of the Determinant ......Page 38
2.2 Properties of Determinants ......Page 42
2.3 Cofactor Expansions ......Page 47
2.4 Laplace's Theorem ......Page 51
2.5 The Binet-Cauchy Formula ......Page 54
2.6 Adjoint and Inverse Matrices ......Page 57
2.7 Elementary Operations on Matrices ......Page 62
2.8 Rank of a Matrix ......Page 68
2.9 Systems of Linear Equations and Matrices ......Page 71
2.10 The LU Decomposition ......Page 76
2.11 Miscellaneous Exercises ......Page 78
3.1 Definition of a Linear Space ......Page 86
3.2 Subspaces ......Page 90
3.3 Linear Combinations ......Page 93
3.4 Linear Dependence and Independence ......Page 95
3.5 The Notion of a Basis ......Page 98
3.6 Sum and Direct Sum of Subspaces ......Page 102
3.7 Matrix Representation and Rank ......Page 106
3.8 Some Properties of Matrices Related to Rank ......Page 110
3.9 Change of Basis and Transition Matrices ......Page 113
3.10 Solution of Equations ......Page 115
3.11 Unitary and Euclidean Spaces ......Page 119
3.12 Orthogonal Systems ......Page 122
3.13 Orthogonal Subspaces ......Page 126
3.14 Miscellaneous Exercises ......Page 128
4 Linear Transformations and Matrices......Page 131
4.1 Linear Transformations ......Page 132
4.2 Matrix Representation of Linear Transformations ......Page 137
4.3 Matrix Representations, Equivalence, and Similarity ......Page 142
4.4 Some Properties of Similar Matrices ......Page 146
4.5 Image and Kernel of a Linear Transformation ......Page 148
4.6 Invertible Transformations ......Page 153
4.7 Restrictions, Invariant Subspaces, and Direct Sums of Transformations ......Page 157
4.8 Direct Sums and Matrices ......Page 160
4.9 Eigenvalues and Eigenvectors of a Transformation ......Page 162
4.10 Eigenvalues and Eigenvectors of a Matrix ......Page 167
4.11 The Characteristic Polynomial ......Page 170
4.12 The Multiplicities of an Eigenvalue ......Page 174
4.13 First Applications to Differential Equations ......Page 176
4.14 Miscellaneous Exercises ......Page 179
5.1 Adjoint Transformations ......Page 183
5.2 Normal Transformations and Matrices ......Page 189
5.3 Hermitian, Skew-Hermitian, and Definite Matrices ......Page 193
5.4 Square Root of a Definite Matrix and Singular Values ......Page 195
5.5 Congruence and the Inertia of a Matrix ......Page 199
5.6 Unitary Matrices ......Page 203
5.7 Polar and Singular-Value Decompositions ......Page 205
5.8 Idempotent Matrices (Projectors) ......Page 209
5.9 Matrices over the Field of Real Numbers ......Page 215
5.10 Bilinear, Quadratic, and Hermitian Forms ......Page 217
5.11 Finding the Canonical Forms ......Page 220
5.12 The Theory of Small Oscillations ......Page 223
5.13 Admissible Pairs of Matrices ......Page 227
5.14 Miscellaneous Exercises ......Page 232
6 The Jordan Canonical Form: A Geometric Approach......Page 235
6.1 Annihilating Polynomials ......Page 236
6.2 Minimal Polynomials ......Page 239
6.3 Generalized Eigenspaces ......Page 244
6.4 The Structure of Generalized Eigenspaces ......Page 247
6.5 The Jordan Theorem ......Page 251
6.6 Parameters of a Jordan Matrix ......Page 254
6.7 The Real Jordan Form ......Page 257
6.8 Miscellaneous Exercises ......Page 259
7.1 The Notion of a Matrix Polynomial ......Page 261
7.2 Division of Matrix Polynomials ......Page 263
7.3 Elementary Operations and Equivalence ......Page 268
7.4 A Canonical Form for a Matrix Polynomial ......Page 271
7.5 Invariant Polynomials and the Smith Canonical Form ......Page 274
7.6 Similarity and the First Normal Form ......Page 277
7.7 Elementary Divisors ......Page 280
7.8 The Second Normal Form and the Jordan Normal Form ......Page 284
7.9 The Characteristic and Minimal Polynomials ......Page 286
7.10 The Smith Form: Differential and Difference Equations ......Page 289
7.11 Miscellaneous Exercises ......Page 293
8 The Variational Method......Page 297
8.1 Field of Values. Extremal Eigenvalues of a Hermitian Matrix ......Page 298
8.2 Courant-Fischer Theory and the Rayleigh Quotient ......Page 301
8.3 The Stationary Property of the Rayleigh Quotient ......Page 304
8.4 Problems with Constraints ......Page 305
8.5 The Rayleigh Theorem and Definite Matrices ......Page 309
8.6 The Jacobi-Gundelfinger-Frobenius Method ......Page 311
8.7 An Application of the Courant-Fischer Theory ......Page 315
8.8 Applications to the Theory of Small Vibrations ......Page 317
9 Functions of Matrices......Page 319
9.1 Functions Defined on the Spectrum of a Matrix ......Page 320
9.2 Interpolatory Polynomials ......Page 321
9.3 Definition of a Function of a Matrix ......Page 323
9.4 Properties of Functions of Matrices ......Page 325
9.5 Spectral Resolution of f(A) ......Page 329
9.6 Component Matrices and Invariant Subspaces ......Page 335
9.7 Further Properties of Functions of Matrices ......Page 337
9 8 Sequences and Series of Matrices ......Page 340
9.9 The Resolvent and the Cauchy Theorem for Matrices ......Page 344
9.10 Applications to Differential Equations ......Page 349
9.11 Observable and Controllable Systems ......Page 355
9.12 Miscellaneous Exercises ......Page 360
10.1 The Notion of a Norm ......Page 365
10.2 A Vector Norm as a Metric: Convergence ......Page 369
10.3 Matrix Norms ......Page 373
10.4 Induced Matrix Norms ......Page 377
10.5 Absolute Vector Norms and Lower Bounds of a Matrix ......Page 382
10.6 The Gerigorin Theorem ......Page 386
10.7 Gerlgorin Disks and Irreducible Matrices ......Page 389
10.8 The Schur Theorem ......Page 392
10.9 Miscellaneous Exercises ......Page 395
11.1 Perturbations in the Solution of Linear Equations ......Page 398
11.2 Perturbations of the Eigenvalues of a Simple Matrix ......Page 402
11.3 Analytic Perturbations ......Page 406
11.4 Perturbation of the Component Matrices ......Page 408
11.5 Perturbation of an Unrepeated Eigenvalue ......Page 410
11.6 Evaluation of the Perturbation Coefficients ......Page 412
11.7 Perturbation of a Multiple Eigenvalue ......Page 414
12.1 The Notion of a Kronecker Product ......Page 421
12.2 Eigenvalues of Kronecker Products and Composite Matrices ......Page 426
12.3 Applications of the Kronecker Product to Matrix Equations ......Page 428
12.4 Commuting Matrices ......Page 431
12.5 Solutions of AX + XB = C ......Page 436
12.6 One-Sided Inverses ......Page 439
12.7 Generalized Inverses ......Page 443
12.8 The Moore-Penrose Inverse ......Page 447
12.9 The Best Approximate Solution of the Equation Ax = b ......Page 450
12.10 Miscellaneous Exercises ......Page 453
13.1 The Lyapunov Stability Theory and Its Extensions ......Page 456
13.2 Stability with Respect to the Unit Circle ......Page 466
13.3 The Bezoutian and the Resultant ......Page 469
13.4 The Hermite and the Routh-Hurwitz Theorems ......Page 476
13.5 The Schur-Cohn Theorem ......Page 481
13.6 Perturbations of a Real Polynomial ......Page 483
13.7 The Li6nard-Chipart Criterion ......Page 485
13.8 The Markov Criterion ......Page 489
13.9 A Determinantal Version of the Routh-Hurwitz Theorem ......Page 493
13.10 The Cauchy Index and Its Applications ......Page 497
14 Matrix Polynomials......Page 504
14.1 Linearization of a Matrix Polynomial ......Page 505
14.2 Standard Triples and Pairs ......Page 508
14.3 The Structure of Jordan Triples ......Page 515
14.4 Applications to Differential Equations ......Page 521
14.5 General Solutions of Differential Equations ......Page 524
14.6 Difference Equations ......Page 527
14.7 A Representation Theorem ......Page 531
14.8 Multiples and Divisors ......Page 533
14.9 Solvents of Monic Matrix Polynomials ......Page 535
15 Nonnegative Matrices......Page 542
15.1 Irreducible Matrices ......Page 543
15.2 Nonnegative Matrices and Nonnegative Inverses ......Page 545
15.3 The Perron-Frobenius Theorem (I) ......Page 547
15.4 The Perron-Frobenius Theorem (II) ......Page 553
15.5 Reducible Matrices ......Page 558
15.6 Primitive and Imprimitive Matrices ......Page 559
15.7 Stochastic Matrices ......Page 556
15.8 Markov Chains ......Page 565
Appendix 1: A Survey of Scalar Polynomials ......Page 568
Appendix 2: Some Theorems and Notions from Analysis ......Page 572
Appendix 3: Suggestions for Further Reading ......Page 575
Index ......Page 578