نظریه ماتریس‌ها (مطالعات تحصیلات تکمیلی در ریاضیات، جلد ۱۴۷)

Matrix theory is a classical topic of algebra that had originated, in its current form, in the middle of the 19th century. It is remarkable that for more than 150 years it continues to be an active area of research full of new discoveries and new applications. This book presents modern perspectives of matrix theory at the level accessible to graduate students. It differs from other books on the subject in several aspects. First, the book treats certain topics that are not found in the standard textbooks, such as completion of partial matrices, sign patterns, applications of matrices in combinatorics, number theory, algebra, geometry, and polynomials. There is an appendix of unsolved problems with their history and current state. Second, there is some new material within traditional topics such as Hopf's eigenvalue bound for positive matrices with a proof, a proof of Horn's theorem on the converse of Weyl's theorem, a proof of Camion-Hoffman's theorem on the converse of the diagonal dominance theorem, and Audenaert's elegant proof of a norm inequality for commutators. Third, by using powerful tools such as the compound matrix and Gröbner bases of an ideal, much more concise and illuminating proofs are given for some previously known results. This makes it easier for the reader to gain basic knowledge in matrix theory and to learn about recent developments. Readership: Graduate students, research mathematicians, and engineers interested in 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) 1. Preliminaries -- 2. Tensor Products And Compound Matrices -- 3. Hermitian Matrices And Majorization -- 4. Singular Values And Unitarily Invariant Norms -- 5. Perturbation Of Matrices -- 6. Nonnegative Matrices -- 7. Completion Of Partial Matrices -- 8. Sign Patterns -- 9. Miscellaneous Topics -- 10. Applications Of Matrices. Xingzhi Zhan. Includes Bibliographical References (pages 237-247) And Index.