![Indefinite Linear Algebra and Applications - Israel Gohberg](https://www.booktopia.com.au/covers/big/9783764373498/0000/indefinite-linear-algebra-and-applications.jpg)
Indefinite Linear Algebra and Applications
By: Israel Gohberg, Peter Lancaster, Leiba Rodman
Paperback | 18 October 2005
At a Glance
357 Pages
23.5 x 15.5 x 1.96
FREE SHIPPING
Paperback
$105.19
or 4 interest-free payments of $26.30 with
orAims to ship in 30 to 35 business days
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
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 |
Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9783764373498
ISBN-10: 3764373490
Published: 18th October 2005
Format: Paperback
Language: English
Number of Pages: 357
Audience: College, Tertiary and University
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5 x 1.96
Weight (kg): 0.52
Shipping
Standard Shipping | Express Shipping | |
---|---|---|
Metro postcodes: | $9.99 | $14.95 |
Regional postcodes: | $9.99 | $14.95 |
Rural postcodes: | $9.99 | $14.95 |
How to return your order
At Booktopia, we offer hassle-free returns in accordance with our returns policy. If you wish to return an item, please get in touch with Booktopia Customer Care.
Additional postage charges may be applicable.
Defective items
If there is a problem with any of the items received for your order then the Booktopia Customer Care team is ready to assist you.
For more info please visit our Help Centre.
You Can Find This Book In
![FIRST LOOK AT RIGOROUS PROB..(2ND ED) : Second Edition - JEFFREY S ROSENTHAL](https://www.booktopia.com.au/covers/200/9789812703712/6572/first-look-at-rigorous-prob-2nd-ed-.jpg)
FREE SHIPPING
RRP $59.99
$53.95
OFF
![2ND STEP MATH OLYMPIAD PROB(V7) : Mathematical Olympiad Series - DEREK HOLTON](https://www.booktopia.com.au/covers/200/9789814327879/4119/2nd-step-math-olympiad-prob-v7-.jpg)
FREE SHIPPING
RRP $68.99
$62.25
OFF
![Developing Thinking in Algebra : Published in Association with The Open University - John Mason](https://www.booktopia.com.au/covers/200/9781412911719/null/developing-thinking-in-algebra.jpg)
FREE SHIPPING
RRP $103.25
$82.75
OFF
![Lec Note Math Olym : Jnr SEC (V2) - Xu Jiagu](https://www.booktopia.com.au/covers/200/9789814293556/null/lec-note-math-olym.jpg)
FREE SHIPPING
Paperback
RRP $45.99
$41.50
OFF
![How to Fall Slower Than Gravity : And Other Everyday (and Not So Everyday) Uses of Mathematics and Physical Reasoning - Paul Nahin](https://www.booktopia.com.au/covers/200/9780691229171/6313/how-to-fall-slower-than-gravity.jpg)
How to Fall Slower Than Gravity
And Other Everyday (and Not So Everyday) Uses of Mathematics and Physical Reasoning
Paperback
RRP $32.99
$24.25
OFF