Indefinite Linear Algebra and Applications

363 Pages·2005·1.721 MB·English

by  Israel Gohberg, Peter Lancaster, Leiba Rodman

Israel Gohberg
Peter Lancaster
Leiba Rodman
Indefinite
Linear Algebra
and
Applications
Birkhäuser
Basel Boston Berlin
• •
Authors:
Israel Gohberg Peter Lancaster
School of Mathematical Sciences Department of Mathematics and Statistics
Raymond and Beverly Sackler University of Calgary
Faculty of Exact Sciences Calgary, Alberta T2N 1N4, Canada
Tel Aviv University e-mail: [e-mail]
Ramat Aviv 69978, Israel
e-mail: [e-mail]
Leiba Rodman
Department of Mathematics
College of William and Mary
P.O. Box 8795
Williamsburg, VA 23187-8795, USA
e-mail: [e-mail]
2000 Mathematics Subject Classification 32-01
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7349-0 Birkhäuser Verlag, Basel – Boston – Berlin
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ISBN-10: 3-7643-7349-0 e-ISBN: 3-7643-7350-4
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We fondly dedicate this book to family members:
Israel Gohberg: To his wife, children, and grandchildren.
Peter Lancaster: To his wife, Diane.
Leiba Rodman: To Ella, Daniel, Ruth, Benjamin, Naomi.
Preface
Thefollowingtopicsofmathematicalanalysishavebeendevelopedinthelastfifty
years:thetheoryoflinearcanonicaldifferentialequationswithperiodicHamiltoni-
ans,the theory of matrix polynomials with selfadjoint coefficients, linear differen-
tial and difference equations of higher order with selfadjoint constant coefficients,
andalgebraicRiccatiequations.Allofthesetheories,andothers,arebasedonrel-
atively recent results of linear algebra in spaces with an indefinite inner product,
i.e., linear algebra in which the usual positive definite inner product is replaced
by an indefinite one. More concisely, we call this subject indefinite linear algebra.
Thisbookhasthestructureofagraduatetextinwhichchaptersofadvanced
linear algebra form the core. The development of our topics follows the lines of
a usuallinear algebracourse.However,chaptersgiving comprehensivetreatments
of differential and difference equations, matrix polynomials and Riccati equations
are interwoven as the necessary techniques are developed.
The main source of material is our earlier monograph in this field: Matrices
and Indefinite Scalar Products, [40]. The present book differs in objectives and
material.Somechaptershavebeenexcluded,othershavebeenadded,andexercises
have been added to all chapters. An appendix is also included. This may serve as
a summary and refresher on standard results as well as a source for some less
familiar material from linear algebra with a definite inner product. The theory
developedhere hasbecome anessentialpartoflinearalgebra.This,togetherwith
the many significantareasofapplication,andthe accessiblestyle,make this book
useful for engineers, scientists and mathematicians alike.
Acknowledgements
The authors gratefully acknowledge support from several projects and organi-
zations: Israel Gohberg acknowledges the generous support of the Silver Family
FoundationandtheSchoolofMathematicalSciencesofTel-AvivUniversity.Peter
Lancaster acknowledges continuing support from the Natural Sciences and Engi-
neeringResearchCouncilofCanada.SupportfromN.J.HighamoftheUniversity
ofManchesterforaresearchfellowshiptenableduringthepreparationofthiswork
viii Preface
is also gratefully acknowledged. Leiba Rodman acknowledges partial support by
NSF grant DMS-9988579, and by the Summer Research Grant and Faculty Re-
search Assignment provided by the College of William and Mary.
Contents
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 OrthogonalBases . . . . . . . . . . . . . . . . . 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 Adjoint Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Dissipative Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.8 Symplectic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Theorem 5.1.1. . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Classification of Matrices by Unitary Similarity . . . . . . . . . . . 82
5.5 Signature Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 Structure of H-Selfadjoint Matrices . . . . . . . . . . . . . . . . . . 89
5.7 H-Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Theorem 6.1.5. . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Comparison with Results in the Complex Case . . . . . . . . . . . 131
6.4 Connected Components of Real Unitary Similarity Classes . . . . . 133
6.5 ConnectedComponents ofRealUnitary SimilarityClasses(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-Selfadjoint Matrices . . . . . . . . . . . . . . . . . 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-NormalLinear Transformationsand Pairsof 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-Selfadjoint Matrices . . . . . . . . . . 179
9.2 Stably Diagonalizable H-Selfadjoint Matrices . . . . . . . . . . . . 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 GeneralPerturbationsandStablyDiagonalizableH-UnitaryMatrices198
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 Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
A.2 The Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
A.3 Riesz Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
A.4 Linear Matrix 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