Matrices : algebra, analysis, and applications

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. Contents Preface 1 Domains, Modules and Matrices 1.1 Rings, Domains and Fields 1.2 Bezout Domains 1.3 DU,DP and DE Domains 1.4 Factorizations in D[x] 1.5 Elementary Divisor Domains 1.6 Modules 1.7 Determinants 1.8 Algebraically Closed Fields 1.9 The Resultant and the Discriminant 1.10 The Ring F[x1, . . . , xn] 1.11 Matrices and Homomorphisms 1.12 Hermite Normal Form 1.13 Systems of Linear Equations over Bezout Domains 1.14 Smith Normal Form 1.15 Local Analytic Functions in One Variable 1.16 The Local–Global Domains in Cp 1.17 Historical Remarks 2 Canonical Forms for Similarity 2.1 Strict Equivalence of Pencils 2.2 Similarity of Matrices 2.3 The Companion Matrix 2.4 Splitting to Invariant Subspaces 2.5 An Upper Triangular Form 2.6 Jordan Canonical Form 2.7 Some Applications of Jordan Canonical Form 2.8 The Matrix Equation AX − XB=0 2.9 A Criterion for Similarity of Two Matrices 2.10 The Matrix Equation AX − XB = C 2.11 A Case of Two Nilpotent Matrices 2.12 Historical Remarks 3 Functions of Matrices and Analytic Similarity 3.1 Components of a Matrix and Functions of Matrices 3.2 Cesaro Convergence of Matrices 3.3 An Iteration Scheme 3.4 Cauchy Integral Formula for Functions of Matrices 3.5 A Canonical Form over HA 3.6 Analytic, Pointwise and Rational Similarity 3.7 A Global Splitting 3.8 First Variation of a Geometrically Simple Eigenvalue 3.9 Analytic Similarity over H0 3.10 Strict Similarity of Matrix Polynomials 3.11 Similarity to Diagonal Matrices 3.12 Property L 3.13 Strict Similarity of Pencils and Analytic Similarity 3.14 Historical Remarks 4 Inner Product Spaces 4.1 Inner Product 4.2 Special Transformations in IPS 4.3 Symmetric Bilinear and Hermitian Forms 4.4 Max–Min Characterizations of Eigenvalues 4.5 Positive Definite Operators and Matrices 4.6 Convexity 4.7 Majorization 4.8 Spectral Functions 4.9 Inequalities for Traces 4.10 Singular Value Decomposition (SVD) 4.11 Characterizations of Singular Values 4.12 Moore–Penrose Generalized Inverse 4.13 Approximation by Low Rank Matrices 4.14 CUR-Approximations 4.15 Some Special Maximal Spectral Problems 4.16 Multiplicity Index of a Subspace of S(V) 4.17 Rellich’s Theorem 4.18 Hermitian Pencils 4.19 Eigenvalues of Sum of Hermitian Matrices 4.20 Perturbation Formulas for Eigenvalues and Eigenvectors of Hermitian Pencils 4.21 Historical Remarks 5 Elements of Multilinear Algebra 5.1 Tensor Product of Two FreeModules 5.2 Tensor Product of Several Free Modules 5.3 Sparse Bases of Subspaces 5.4 Tensor Products of Inner Product Spaces 5.5 Matrix Exponents 5.6 Historical Remarks 6 Non-Negative Matrices 6.1 Graphs 6.1.1 Undirected graphs 6.1.2 Directed graphs 6.1.3 Multigraphs and multidigraphs 6.1.4 Matrices and graphs 6.2 Perron–Frobenius Theorem 6.3 Index of Primitivity 6.4 Reducible Matrices 6.5 Stochastic Matrices and Markov Chains 6.6 Friedland–Karlin Results 6.7 Log-Convexity 6.8 Min–Max Characterizations of ρ(A) 6.9 An Application to Cellular Communication 6.9.1 Introduction 6.9.2 Statement of problems 6.9.3 Relaxations of optimal problems 6.9.4 Preliminary results 6.9.5 Reformulation of optimal problems 6.9.6 Algorithms for sum rate maximization 6.10 Historical Remarks 7 Various Topics 7.1 Norms over Vector Spaces 7.2 Numerical Ranges and Radii 7.3 Superstable Norms 7.4 Operator Norms 7.5 Tensor Products of Convex Sets 7.6 The Complexity of conv Ωn Ωm 7.7 Variation of Tensor Powers and Spectra 7.8 Variation of Permanents 7.9 Vivanti–Pringsheim Theorem and Applications 7.10 Inverse Eigenvalue Problem for Non-Negative Matrices 7.11 Cones 7.12 Historical Remarks Bibliography Index of Symbols Index "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 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"--Back cover.