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Indefinite Linear Algebra and Applications

Publisher: 
Birkhäuser
Number of Pages: 
357
Price: 
59.95
ISBN: 
3-7643-7349-0
Date Received: 
Tuesday, January 24, 2006
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Israel Gohberg, Peter Lancaster, and Leiba Rodman
Publication Date: 
2005
Format: 
Paperback
Category: 
Monograph

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

Publish Book: 
Modify Date: 
Wednesday, May 24, 2006

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