Matrices: Algebra, Analysis And Applications

Matrices: Algebra, Analysis And Applications

by Shmuel Friedland

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Overview

This volume deals with advanced topics in matrix theory using the notions and tools from algebra, analysis, geometry and numerical analysis. It consists of seven chapters that are loosely connected and interdependent. The choice of the topics is very personal and reflects the subjects that the author was actively working on in the last 40 years. Many results appear for the first time in the volume. Readers will encounter various properties of matrices with entries in integral domains, canonical forms for similarity, and notions of analytic, pointwise and rational similarity of matrices with entries which are locally analytic functions in one variable. This volume is also devoted to various properties of operators in inner product space, with tensor products and other concepts in multilinear algebra, and the theory of non-negative matrices. It will be of great use to graduate students and researchers working in pure and applied mathematics, bioinformatics, computer science, engineering, operations research, physics and statistics.

Product Details

ISBN-13: 9789813141032
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 10/30/2015
Pages: 596
Product dimensions: 6.00(w) x 9.00(h) x 1.30(d)

Table of Contents

Preface vii

1 Domains, Modules and Matrices 1

1.1 Rings, Domains and Fields 1

1.2 Bezout Domains 4

1.3 Du, Dpp and DeDomains 9

1.4 Factorizations in D[x] 12

1.5 Elementary Divisor Domains 16

1.6 Modules 17

1.7 Determinants 25

1.8 Algebraically Closed Fields 30

1.9 The Resultant and the Discriminant 32

1.10 The Ring F[x1,…, xn] 37

1.11 Matrices and Homomorphisms 38

1.12 Hermite Normal Form 42

1.13 Systems of Linear Equations over Bezout Domains 51

1.14 Smith Normal Form 55

1.15 Local Analytic Functions in One Variable 60

1.16 The Local-Global Domains in Cp 67

1.17 Historical Remarks 74

2 Canonical Forms for Similarity 75

2.1 Strict Equivalence of Pencils 75

2.2 Similarity of Matrices 82

2.3 The Companion Matrix 84

2.4 Splitting to Invariant Subspaces 87

2.5 An Upper Triangular Form 92

2.6 Jordan Canonical Form 96

2.7 Some Applications of Jordan Canonical Form 99

2.8 The Matrix Equation AX - XB = 0 103

2.9 A Criterion for Similarity of Two Matrices 109

2.10 The Matrix Equation AX - XB = C 114

2.11 A Case of Two Nilpotent Matrices 119

2.12 Historical Remarks 122

3 Functions of Matrices and Analytic Similarity 123

3.1 Components of a Matrix and Functions of Matrices 123

3.2 Cesaro Convergence of Matrices 127

3.3 An Iteration Scheme 131

3.4 Cauchy Integral Formula for Functions of Matrices 133

3.5 A Canonical Form over 143

3.6 Analytic, Pointwise and Rational Similarity 150

3.7 A Global Splitting 154

3.8 First Variation of a Geometrically Simple Eigenvalue 155

3.9 Analytic Similarity over H0 157

3.10 Strict Similarity of Matrix Polynomials 166

3.11 Similarity to Diagonal Matrices 169

3.12 Property L 179

3.13 Strict Similarity of Pencils and Analytic Similarity 185

3.14 Historical Remarks 192

4 Inner Product Spaces 195

4.1 Inner Product 195

4.2 Special Transformations in IPS 200

4.3 Symmetric Bilinear and Hermitian Forms 209

4.4 Max-Min Characterizations of Eigenvalues 212

4.5 Positive Definite Operators and Matrices 221

4.6 Convexity 227

4.7 Majorization 237

4.8 Spectral Functions 245

4.9 Inequalities for Traces 253

4.10 Singular Value Decomposition (SVD) 256

4.11 Characterizations of Singular Values 263

4.12 Moore-Penrose Generalized Inverse 273

4.13 Approximation by Low Rank Matrices 276

4.14 CUR-Approximations 281

4.15 Some Special Maximal Spectral Problems 285

4.16 Multiplicity Index of a Subspace of S(V) 292

4.17 Rellich's Theorem 300

4.18 Hermitian Pencils 304

4.19 Eigenvalues of Sum of Hermitian Matrices 316

4.20 Perturbation Formulas for Eigenvalues and Eigenvectors of Hermitian Pencils 319

4.21 Historical Remarks 323

5 Elements of Multilinear Algebra 325

5.1 Tensor Product of Two Free Modules 325

5.2 Tensor Product of Several Free Modules 333

5.3 Sparse Bases of Subspaces 344

5.4 Tensor Products of Inner Product Spaces 360

5.5 Matrix Exponents 374

5.6 Historical Remarks 392

6 Non-Negative Matrices 393

6.1 Graphs 393

6.1.1 Undirected graphs 393

6.1.2 Directed graphs 394

6.1.3 Mnltigraphs and multidigraphs 395

6.1.4 Matrices and graphs 399

6.2 Perron-Frobenius Theorem 403

6.3 Index of Primitivity 416

6.4 Reducible Matrices 418

6.5 Stochastic Matrices and Markov Chains 427

6.6 Friedland-Karlin Results 437

6.7 Log-Convexity 447

6.8 Min-Max Characterizations of ρ(A) 450

6.9 An Application to Cellular Communication 456

6.9.1 Introduction 456

6.9.2 Statement of problems 457

6.9.3 Relaxations of optimal problems 459

6.9.4 Preliminary results 460

6.9.5 Reformulation of optimal problems 464

6.9.6 Algorithms for sum rate maximization 467

6.10 Historical Remarks 472

7 Various Topics 473

7.1 Norms over Vector Spaces 473

7.2 Numerical Ranges and Radii 482

7.3 Superstable Norms 490

7.4 Operator Norms 494

7.5 Tensor Products of Convex Sets 498

7.6 The Complexity of conv Ωn $$$ Ωm 508

7.7 Variation of Tensor Powers and Spectra 511

7.8 Variation of Permanents 521

7.9 Vivanti-Pringsheim Theorem and Applications 525

7.10 Inverse Eigenvalue Problem for Non-Negative Matrices 533

7.11 Cones 542

7.12 Historical Remarks 554

Bibliography 555

Index of Symbols 571

Index 579

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