Matrix theory

Cover

S Title

Matrix Theory

Copyright
     2013 by the American Mathematical Society
     ISBN 978-0-8218-9491-0
     QA 188. Z43 2013 512.9'434-dc23
     LCCN 2013001353

Contents

Preface

Chapter 1  Preliminaries

     1.1. Classes of Special Matrices

     1.2. The Characteristic Polynomial

     1.3. The Spectral Mapping Theorem

     1.4. Eigenvalues and Diagonal Entries

     1.5. Norms

     1.6. Convergence of the Power Sequence of a Matrix

     1.7. Matrix Decompositions

     1.8. Numerical Range

     1.9. The Companion Matrix of a Polynomial

     1.10. Generalized Inverses

     1.11. Schur Complements

     1.12. Applications of Topological Ideas

     1.13. Grobner Bases

     1.14. Systems of Linear Inequalities

     1.15. Orthogonal Projections and Reducing Subspaces

     1.16. Books and Journals about Matrices

     Exercises

Chapter 2  Tensor Products and Compound Matrices

     2.1. Definitions and Basic Properties

     2.2. Linear Matrix Equations

     2.3. Frobenius-Konig Theorem

     2.4. Compound Matrices

     Exercises

Chapter 3  Hermitian Matrices and Majorization

     3.1. Eigenvalues of Hermitian Matrices

     3.2. Majorization and Doubly Stochastic Matrices

     3.3. Inequalities for Positive Semidefinite Matrices

     Exercises

Chapter 4  Singular Values and Unitarily Invariant Norms

     4.1. Singular Values

     4.2. Symmetric Gauge Functions

     4.3. Unitarily Invariant Norms

     4.4. The Cartesian Decomposition of Matrices

     Exercises

Chapter 5  Perturbation of Matrices

     5.1. Eigenvalues

     5.2. The Polar Decomposition

     5.3. Norm Estimation of Band Parts

     5.4. Backward Perturbation Analysis

     Exercises

Chapter 6  Nonnegative Matrices

     6.1. Perron-Frobenius Theory

     6.2. Matrices and Digraphs

     6.3. Primitive and Imprimitive Matrices

     6.4. Special Classes of Nonnegative Matrices

     6.5. Two Theorems about Positive Matrices

     Exercises

Chapter 7  Completion of Partial Matrices

     7.1. Friedland's Theorem about Diagonal Completions

     7.2. Farahat-Ledermann's Theorem about Borderline Completions

     7.3. Parrott's Theorem about Norm-Preserving Completions

     7.4. Positive Definite Completions

Chapter 8  Sign Patterns

     8.1. Sign-Nonsingular Patterns

     8.2. Eigenvalues

     8.3. Sign Semi-Stable Patterns

     8.4. Sign Patterns Allowing a Positive Inverse

     Exercises

Chapter 9  Miscellaneous Topics

     9.1. Similarity of Real Matrices via Complex Matrices

     9.2. Inverses of Band Matrices

     9.3. Norm Bounds for Commutators

     9.4. The Converse of the Diagonal Dominance Theorem

     9.5. The Shape of the Numerical Range

     9.6. An Inversion Algorithm

     9.7. Canonical Forms for Similarity

     9.8. Extremal Sparsity of the Jordan Canonical Form

Chapter 10  Applications of Matrices

     10.1. Combinatorics

     10.2. Number Theory

     10.3. Algebra

     10.4. Geometry

     10.5. Polynomials

Unsolved Problems

     1. Existence of Hadamard matrices

     2. Characterization of the eigenvalues of nonnegative matrices

     3. The permanental dominance conjecture

     4. The Marcus-de Oliveira conjecture

     5. Permanents of Hadamard matrices

     6. The S-matrix conjecture

     7. The Grone-Merris conjecture on Laplacian spectra

     8. The CP-rank conjecture

     9. Singular value inequalities

     10. Expressing real matrices as linear combinations of orthogonal matrices

     11. The 2n conjecture on spectrally arbitrary sign patterns

     12. Sign patterns of nonnegative matrices

     13. Eigenvalues of real symmetric matrices

     14. Sharp constants in spectral variation

     15. Powers of 0-1 matrices

Bibliography

Notation

Index

Back Cover

© 2014 MicrosoftTermsPrivacyDevelopersEnglish (United States)