Preface vii
1 Introduction and Outline 1
1.1 Description of the Contents . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . 3
2 Indefinite Inner Products 7
2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Orthogonality and Orthogonal Bases . . . . . . . . . . . . . . . . . 9
2.3 Classification of Subspaces . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Orthogonalization and Orthogonal Polynomials 19
3.1 Regular Orthogonalizations . . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Theorems of Szeg˝o and Krein . . . . . . . . . . . . . . . . . . 27
3.3 One-Step Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Determinants of One-Step Completions . . . . . . . . . . . . . . . 36
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Classes of Linear Transformations 45
4.1 AdjointMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 H-Selfadjoint Matrices: Examples and Simplest Properties . . . . . 48
4.3 H-Unitary Matrices: Examples and Simplest Properties . . . . . . 50
4.4 A Second Characterization of H-Unitary Matrices . . . . . . . . . 54
4.5 Unitary Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 DissipativeMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.8 SymplecticMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
x Contents
5 Canonical Forms 73
5.1 Description of a Canonical Form . . . . . . . . . . . . . . . . . . . 73
5.2 First Application of the Canonical Form . . . . . . . . . . . . . . . 75
5.3 Proof of Theorem5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Classification of Matrices by Unitary Similarity . . . . . . . . . . . 82
5.5 SignatureMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Structure of H-SelfadjointMatrices . . . . . . . . . . . . . . . . . . 89
5.7 H-DefiniteMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.8 Second Description of the Sign Characteristic . . . . . . . . . . . . 92
5.9 Stability of the Sign Characteristic . . . . . . . . . . . . . . . . . . 95
5.10 Canonical Forms for Pairs of Hermitian Matrices . . . . . . . . . . 96
5.11 Third Description of the Sign Characteristic . . . . . . . . . . . . . 98
5.12 Invariant Maximal Nonnegative Subspaces . . . . . . . . . . . . . . 99
5.13 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.14 Canonical Forms for H-Unitaries: First Examples . . . . . . . . . 107
5.15 Canonical Forms for H-Unitaries: General Case . . . . . . . . . . . 110
5.16 First Applications of the Canonical Form of H-Unitaries . . . . . . 118
5.17 Further Deductions from the Canonical Form . . . . . . . . . . . . 119
5.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.19 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6 Real H-Selfadjoint Matrices 125
6.1 Real H-Selfadjoint Matrices and Canonical Forms . . . . . . . . . 125
6.2 Proof of Theorem6.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Comparison with Results in the Complex Case . . . . . . . . . . . 131
6.4 Connected Components of Real Unitary Similarity Classes . . . . . 133
6.5 Connected Components of Real Unitary Similarity Classes (H Fixed)137
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7 Functions of H-Selfadjoint Matrices 143
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . 145
7.3 Functions of H-SelfadjointMatrices . . . . . . . . . . . . . . . . . 147
7.4 The Canonical Form and Sign Characteristic . . . . . . . . . . . . 150
7.5 Functions which are Selfadjoint in another Indefinite Inner Product 154
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 H-Normal Matrices 159
8.1 Decomposability: First Remarks . . . . . . . . . . . . . . . . . . . 159
8.2 H-Normal Linear Transformations and Pairs of Commuting Matrices163
8.3 On Unitary Similarity in an Indefinite Inner Product . . . . . . . . 165
8.4 The Case of Only One Negative Eigenvalue of H . . . . . . . . . . 166
Contents xi
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9 General Perturbations. Stability of Diagonalizable Matrices 179
9.1 General Perturbations of H-SelfadjointMatrices . . . . . . . . . . 179
9.2 Stably Diagonalizable H-SelfadjointMatrices . . . . . . . . . . . . 183
9.3 Analytic Perturbations and Eigenvalues . . . . . . . . . . . . . . . 185
9.4 Analytic Perturbations and Eigenvectors . . . . . . . . . . . . . . . 189
9.5 The Real Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.6 Positive Perturbations of H-Selfadjoint Matrices . . . . . . . . . . 193
9.7 H-Selfadjoint Stably r-Diagonalizable Matrices . . . . . . . . . . . 195
9.8 General Perturbations and Stably Diagonalizable H-Unitary Matrices198
9.9 H-Unitarily Stably u-Diagonalizable Matrices . . . . . . . . . . . . 200
9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10 Definite Invariant Subspaces 207
10.1 Semidefinite and Neutral Subspaces: A Particular H . . . . . . . . 207
10.2 Plus Matrices and Invariant Nonnegative Subspaces . . . . . . . . 212
10.3 Deductions from Theorem 10.2.4 . . . . . . . . . . . . . . . . . . . 217
10.4 Expansive, Contractive Matrices and Spectral Properties . . . . . . 221
10.5 The Real Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11 Differential Equations of First Order 229
11.1 Boundedness of solutions . . . . . . . . . . . . . . . . . . . . . . . 229
11.2 Hamiltonian Systems of Positive Type with Constant Coefficients . 232
11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
12 Matrix Polynomials 237
12.1 Standard Pairs and Triples . . . . . . . . . . . . . . . . . . . . . . 238
12.2 Matrix Polynomials with Hermitian Coefficients . . . . . . . . . . . 242
12.3 Factorization of Hermitian Matrix Polynomials . . . . . . . . . . . 245
12.4 The Sign Characteristic of Hermitian Matrix Polynomials . . . . . 249
12.5 The Sign Characteristic of Hermitian Analytic Matrix Functions . 256
12.6 Hermitian Matrix Polynomials on the Unit Circle . . . . . . . . . . 261
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
12.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
xii Contents
13 Differential and Difference Equations of Higher Order 267
13.1 General Solution of a System of Differential Equations . . . . . . . 267
13.2 Boundedness for a System of Differential Equations . . . . . . . . . 268
13.3 Stable Boundedness for Differential Equations . . . . . . . . . . . . 270
13.4 The Strongly Hyperbolic Case . . . . . . . . . . . . . . . . . . . . . 273
13.5 Connected Components of Differential Equations . . . . . . . . . . 274
13.6 A Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
13.7 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 278
13.8 Stable Boundedness for Difference Equations . . . . . . . . . . . . 281
13.9 Connected Components of Difference Equations . . . . . . . . . . . 284
13.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
13.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
14 Algebraic Riccati Equations 289
14.1 Matrix Pairs in Systems Theory and Control . . . . . . . . . . . . 290
14.2 Origins in Systems Theory . . . . . . . . . . . . . . . . . . . . . . . 293
14.3 Preliminaries on the Riccati Equation . . . . . . . . . . . . . . . . 295
14.4 Solutions and Invariant Subspaces . . . . . . . . . . . . . . . . . . 296
14.5 Symmetric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 297
14.6 An Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 298
14.7 Existence when M has Real Eigenvalues . . . . . . . . . . . . . . . 303
14.8 Description of Hermitian Solutions . . . . . . . . . . . . . . . . . . 307
14.9 Extremal Hermitian Solutions . . . . . . . . . . . . . . . . . . . . . 309
14.10The CARE with Real Coefficients . . . . . . . . . . . . . . . . . . . 312
14.11The Concerns of Numerical Analysis . . . . . . . . . . . . . . . . . 315
14.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
14.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
A Topics from Linear Algebra 319
A.1 HermitianMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
A.2 The Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
A.3 Riesz Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
A.4 LinearMatrix Equations . . . . . . . . . . . . . . . . . . . . . . . . 335
A.5 Perturbation Theory of Subspaces . . . . . . . . . . . . . . . . . . 335
A.6 Diagonal Forms for Matrix Polynomials and Matrix Functions . . . 338
A.7 Convexity of the Numerical Range . . . . . . . . . . . . . . . . . . 342
A.8 The Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . 344
A.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Bibliography 349
Index 355