Indefinite Linear Algebra and Applications

By: Israel Gohberg, Peter Lancaster, Leiba Rodman

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Paperback | 18 October 2005

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372 Pages

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This book covers recent results in linear algebra with indefinite inner product. It includes applications to differential and difference equations with symmetries, matrix polynomials and Riccati equations. These applications are based on linear algebra in spaces with indefinite inner product. The latter forms an independent branch of linear algebra called indefinite linear algebra. This new subject is presented following the principles of a standard linear algebra course.

Industry Reviews

This is a splendidly written book, one of a number by three of the world's leading linear algebraists, that collects many results on indefinite linear algebra from the literature and puts them all in a single place. The book is based in part on a 1983 book titled Matrices and Indefinite Scalar Products by the same authors and publisher but includes many contemporary results from the journal literature and elsewhere. In fact, the ends of many of the chapters provide a guide to sources where further materials are to be found. There are some excellent exercises collected at the end of each chapter making the book ideal as a graduate-level textbook.

(SIAM Review)

"The reviewer is convinced that the present book will be welcome by all readers interested in applications of indefinite linear algebra in matrix polynomials, differential equations and difference equations with constant coefficients." Mircea Crasmareanu, Analele Stiintifice

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Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Complex Analysis Functional Analysis & Transforms Differential Calculus & Equations

Preface	p. vii
Introduction and Outline	p. 1
Description of the Contents	p. 2
Notation and Conventions	p. 3
Indefinite Inner Products	p. 7
Definition	p. 7
Orthogonality and Orthogonal Bases	p. 9
Classification of Subspaces	p. 11
Exercises	p. 14
Notes	p. 18
Orthogonalization and Orthogonal Polynomials	p. 19
Regular Orthogonalizations	p. 19
The Theorems of Szego and Krein	p. 27
One-Step Theorem	p. 29
Determinants of One-Step Completions	p. 36
Exercises	p. 40
Notes	p. 44
Classes of Linear Transformations	p. 45
Adjoint Matrices	p. 45
H-Selfadjoint Matrices: Examples and Simplest Properties	p. 48
H-Unitary Matrices: Examples and Simplest Properties	p. 50
A Second Characterization of H-Unitary Matrices	p. 54
Unitary Similarity	p. 55
Contractions	p. 57
Dissipative Matrices	p. 59
Symplectic Matrices	p. 62
Exercises	p. 66
Notes	p. 72
Canonical Forms	p. 73
Description of a Canonical Form	p. 73
First Application of the Canonical Form	p. 75
Proof of Theorem 5.1.1	p. 77
Classification of Matrices by Unitary Similarity	p. 82
Signature Matrices	p. 85
Structure of H-Selfadjoint Matrices	p. 89
H-Definite Matrices	p. 91
Second Description of the Sign Characteristic	p. 92
Stability of the Sign Characteristic	p. 95
Canonical Forms for Pairs of Hermitian Matrices	p. 96
Third Description of the Sign Characteristic	p. 98
Invariant Maximal Nonnegative Subspaces	p. 99
Inverse Problems	p. 106
Canonical Forms for H-Unitaries: First Examples	p. 107
Canonical Forms for H-Unitaries: General Case	p. 110
First Applications of the Canonical Form of H-Unitaries	p. 118
Further Deductions from the Canonical Form	p. 119
Exercises	p. 120
Notes	p. 123
Real H-Selfadjoint Matrices	p. 125
Real H-Selfadjoint Matrices and Canonical Forms	p. 125
Proof of Theorem 6.1.5	p. 128
Comparison with Results in the Complex Case	p. 131
Connected Components of Real Unitary Similarity Classes	p. 133
Connected Components of Real Unitary Similarity Classes (H Fixed)	p. 137
Exercises	p. 140
Notes	p. 142
Functions of H-Selfadjoint Matrices	p. 143
Preliminaries	p. 143
Exponential and Logarithmic Functions	p. 145
Functions of H-Selfadjoint Matrices	p. 147
The Canonical Form and Sign Characteristic	p. 150
Functions which are Selfadjoint in another Indefinite Inner Product	p. 154
Exercises	p. 156
Notes	p. 158
H-Normal Matrices	p. 159
Decomposability: First Remarks	p. 159
H-Normal Linear Transformations and Pairs of Commuting Matrices	p. 163
On Unitary Similarity in an Indefinite Inner Product	p. 165
The Case of Only One Negative Eigenvalue of H	p. 166
Exercises	p. 174
Notes	p. 177
General Perturbations. Stability of Diagonalizable Matrices	p. 179
General Perturbations of H-Selfadjoint Matrices	p. 179
Stably Diagonalizable H-Selfadjoint Matrices	p. 183
Analytic Perturbations and Eigenvalues	p. 185
Analytic Perturbations and Eigenvectors	p. 189
The Real Case	p. 192
Positive Perturbations of H-Selfadjoint Matrices	p. 193
H-Selfadjoint Stably r-Diagonalizable Matrices	p. 195
General Perturbations and Stably Diagonalizable H-Unitary Matrices	p. 198
H-Unitarily Stably u-Diagonalizable Matrices	p. 200
Exercises	p. 203
Notes	p. 205
Definite Invariant Subspaces	p. 207
Semidefinite and Neutral Subspaces: A Particular H	p. 207
Plus Matrices and Invariant Nonnegative Subspaces	p. 212
Deductions from Theorem 10.2.4	p. 217
Expansive, Contractive Matrices and Spectral Properties	p. 221
The Real Case	p. 226
Exercises	p. 227
Notes	p. 228
Differential Equations of First Order	p. 229
Boundedness of solutions	p. 229
Hamiltonian Systems of Positive Type with Constant Coefficients	p. 232
Exercises	p. 234
Notes	p. 236
Matrix Polynomials	p. 237
Standard Pairs and Triples	p. 238
Matrix Polynomials with Hermitian Coefficients	p. 242
Factorization of Hermitian Matrix Polynomials	p. 245
The Sign Characteristic of Hermitian Matrix Polynomials	p. 249
The Sign Characteristic of Hermitian Analytic Matrix Functions	p. 256
Hermitian Matrix Polynomials on the Unit Circle	p. 261
Exercises	p. 263
Notes	p. 266
Differential and Difference Equations of Higher Order	p. 267
General Solution of a System of Differential Equations	p. 267
Boundedness for a System of Differential Equations	p. 268
Stable Boundedness for Differential Equations	p. 270
The Strongly Hyperbolic Case	p. 273
Connected Components of Differential Equations	p. 274
A Special Case	p. 276
Difference Equations	p. 278
Stable Boundedness for Difference Equations	p. 281
Connected Components of Difference Equations	p. 284
Exercises	p. 286
Notes	p. 288
Algebraic Riccati Equations	p. 289
Matrix Pairs in Systems Theory and Control	p. 290
Origins in Systems Theory	p. 293
Preliminaries on the Riccati Equation	p. 295
Solutions and Invariant Subspaces	p. 296
Symmetric Equations	p. 297
An Existence Theorem	p. 298
Existence when M has Real Eigenvalues	p. 303
Description of Hermitian Solutions	p. 307
Extremal Hermitian Solutions	p. 309
The CARE with Real Coefficients	p. 312
The Concerns of Numerical Analysis	p. 315
Exercises	p. 317
Notes	p. 318
Topics from Linear Algebra	p. 319
Hermitian Matrices	p. 319
The Jordan Form	p. 321
Riesz Projections	p. 332
Linear Matrix Equations	p. 335
Perturbation Theory of Subspaces	p. 335
Diagonal Forms for Matrix Polynomials and Matrix Functions	p. 338
Convexity of the Numerical Range	p. 342
The Fixed Point Theorem	p. 344
Exercises	p. 345
Bibliography	p. 349
Index	p. 355
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Indefinite Linear Algebra and Applications

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