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Matrix polynomials

Author: Gohberg, I. ; Lancaster, P. ; Rodman, L. Series: Classics in applied mathematics Publisher: Society for Industrial and Applied Mathematics (SIAM) 2009.Edition: 2nd ed.Language: EnglishDescription: 409 p. ; 23 cm.ISBN: 9780898716818Type of document: BookBibliography/Index: Includes bibliographical references and index
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Book Europe Campus
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Print QA155 .G64 2009
(Browse shelf)
001274486
Available 001274486
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Includes bibliographical references and index

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Matrix Polynomials Table of contents Preface to the Classics Edition Preface Errata xv xix xxi Introduction Part I MONIC MATRIX POLYNOMIALS Chapter 1 Linearization and Standard Pairs 1.1 1.2 Linearization Application to Differential and Difference Equations 11 15 20 23 29 32 37 40 43 46 9 1.3 The Inverse Problem for Linearization 1.4 Jordan Chains and Solutions of Differential Equations 1.5 Root Polynomials 1.6 1.7 1.8 1.9 1.10 Canonical Set of Jordan Chains Jordan Chains and the Singular Part of the Laurent Expansion Definition of a Jordan Pair of a Monic Polynomial Properties of a Jordan Pair Standard Pairs of a Monic Matrix Polynomial Chapter 2 Representation of Monic Matrix Polynomials 2.1 2.2 2.3 2.4 2.5 2.6 Standard and Jordan Triples Representations of a Monic Matrix Polynomial Resolvent Form and Linear Systems Theory Initial Value Problems and Two-Point Boundary Value Problems Complete Pairs and Second-Order Differential Equations Initial Value Problem for Difference Equations, and the Generalized Newton Identities 50 57 66 70 75 79 Chapter 3 Multiplication and Divisibility 3.1 3.2 3.3 3.4 3.5 3.6 3.7 A Multiplication Theorem Division Process Characterization of Divisors and Supporting Subspaces Example Description of the Quotient and Left Divisors Divisors and Supporting Subspaces for the Adjoint Polynomial Decomposition into a Product of Linear Factors 85 89 96 100 104 111 112 Chapter 4 Spectral Divisors and Canonical Factorization 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Spectral Divisors Linear Divisors and Matrix Equations Stable and Exponentially Growing Solutions of Differential Equations Left and Right Spectral Divisors Canonical Factorization Theorems on Two-Sided Canonical Factorization Wiener-Hopf Factorization for Matrix Polynomials 116 125 129 131 133 139 142 Chapter 5 Perturbation and Stability of Divisors 5.1 5.2 5.3 5.4 5.5 5.6 5.7 The Continuous Dependence of Supporting Subspaces and Divisors Spectral Divisors: Continuous and Analytic Dependence Stable Factorizations Global Analytic Perturbations: Preliminaries Polynomial Dependence Analytic Divisors Isolated and Nonisolated Divisors 147 150 152 155 158 162 164 Chapter 6 Extension Problems 6.1 6.2 6.3 Statement of the Problems and Examples Extensions via Left Inverses Special Extensions 167 169 173 Part II NONMONIC MATRIX POLYNOMIALS Chapter 7 Spectral Properties and Representations 7.1 7.2 7.3 7.4 7.5 The Spectral Data (Finite and Infinite) Linearizations Decomposable Pairs Properties of Decomposable Pairs Decomposable Linearization and a Resolvent Form 181 183 186 188 191 195 7.6 7.7 7.8 Representation and the Inverse Problem 197 201 206 208 211 213 Divisibility of Matrix Polynomials Representation Theorems for Comonic Matrix Polynomials 7.9 Comonic Polynomials from Finite Spectral Data 7.10 Description of Divisors via Invariant Subspaces 7.11 Construction of a Comonic Matrix Polynomial via a Special Generalized Inverse Chapter 8 Applications to Differential and Difference Equations 8.1 8.2 8.3 Differential Equations in the Nonmonic Case Difference Equations in the Nonmonic Case Construction of Differential and Difference Equations with Given Solutions 219 225 227 Chapter 9 Least Common Multiples and Greatest Common Divisors of Matrix Polynomials 9.1 Common Extensions of Admissible Pairs 9.2 Common Restrictions of Admissible Pairs 9.3 Construction of I.c.m. and g.c.d. via Spectral Data 9.4 Vandermonde Matrix and Least Common Multiples 9.5 Common Multiples for Monic Polynomials 9.6 Resultant Matrices and Greatest Common Divisors 232 235 239 240 244 246 253 Part III SELF-ADJOINT MATRIX POLYNOMIALS Chapter 10 General Theory 10.1 10.2 10.3 10.4 10.5 10.6 Simplest Properties Self-Adjoint Triples: Definition Self-Adjoint Triples: Existence Self-Adjoint Triples for Real Self-Adjoint Matrix Polynomials Sign Characteristic of a Self-Adjoint Matrix Polynomial Numerical Range and Eigenvalues 255 260 263 266 273 276 Chapter 11 Factorization of Self-Adjoint Matrix Polynomials 11.1 Symmetric Factorization 11.2 Main Theorem 11.3 Proof of the Main Theorem 11.4 Discussion and Further Deductions 278 279 282 285 Chapter 12 Further Analysis of the Sign Characteristic 12.1 Localization of the Sign Characteristic 12.2 Stability of the Sign Characteristic 290 293 12.3 12.4 12.5 A Sign Characteristic for Self-Adjoint Analytic Matrix Functions Third Description of the Sign Characteristic Nonnegative Matrix Polynomials 294 298 301 Chapter 13 Quadratic Self-Adjoint Polynomials 13.1 13.2 Overdamped Case Weakly Damped Case 305 309 Part IV SUPPLEMENTARY CHAPTERS IN LINEAR ALGEBRA Chapter S1 The Smith Form and Related Problems S1.1 S1.2 S1.3 S1.4 S1.5 S1.6 S1.7 S1.8 The Smith Form Invariant Polynomials and Elementary Divisors Application to Differential Equations with Constant Coefficients Application to Difference Equations Local Smith Form and Partial Multiplicities Equivalence of Matrix Polynomials Jordan Normal Form Functions of Matrices 311 313 319 321 325 330 333 336 337 Chapter S2 The Matrix Equation AX - XB = C S2.1 S2.2 Existence of Solutions of AX -- XB = C Commuting Matrices 342 345 348 Chapter S3 One-Sided and Generalized Inverses Chapter S4 Stable Invariant Subspaces S4.1 S4.2 S4.3 S4.4 S4.5 S4.6 S4.7 Projectors and Subspaces Spectral Invariant Subspaces and Riesz Projectors The Gap between Subspaces The Metric Space of Subspaces Stable Invariant Subspaces: Definition and Main Result Case of a Single Eigenvalue General Case of Stable Invariant Subspaces 353 356 360 363 366 367 369 Chapter S5 Indefinite Scalar Product Spaces S5.1 S5.2 Canonical Form of a Self-Adjoint Matrix and the Indefinite Scalar Product Proof of Theorem S5.1 376 378 S5.3 S5.4 Uniqueness of the Sign Characteristic Second Description of the Sign Characteristic 383 386 Chapter S6 Analytic Matrix Functions S6.1 S6.2 General Results Analytic Perturbations of Self-Adjoint Matrices 388 394 References List of Notations and Conventions Index 397 403 405

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