Matrix Inequalities for Iterative Systems: 1st Edition (Hardback) book cover

Matrix Inequalities for Iterative Systems

1st Edition

By Hanjo Taubig

CRC Press

218 pages | 3 B/W Illus.

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Hardback: 9781498777773
pub: 2016-11-18
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Description

The book reviews inequalities for weighted entry sums of matrix powers. Applications range from mathematics and CS to pure sciences. It unifies and generalizes several results for products and powers of sesquilinear forms derived from powers of Hermitian, positive-semidefinite, as well as nonnegative matrices. It shows that some inequalities are valid only in specific cases. How to translate the Hermitian matrix results into results for alternating powers of general rectangular matrices? Inequalities that compare the powers of the row and column sums to the row and column sums of the matrix powers are refined for nonnegative matrices. Lastly, eigenvalue bounds and derive results for iterated kernels are improved.

Table of Contents

INTRODUCTION

Notation and Basic Facts

Matrices and Vectors

Graphs

Number of Walks in Graphs

Entry Sums and Matrix Powers

Subsets, Submatrices, and Weighted Entry Sums

Elementary Inequalities for Vectors and Sequences

Motivation

Simple Combinatorial Problems

Automata and Formal Languages

Graph Density, Maximum Clique and Densest k-Subgraph

The Number of Paths and Extremal Graph Theory

Means, Variances, and Irregularity

Random Walks and Markov Chains

Largest Eigenvalue

Population Genetics and Evolutionary Theory

Theoretical Chemistry

Iterated Line Digraphs

Other Applications

Diagonalization and Spectral Decomposition

Relevant Literature

Similar and Diagonalizable Matrices

Hermitian and Real Symmetric Matrices

Adjacency Matrices and Number of Walks in Graphs

Quadratic Forms for Diagonalizable Matrices

UNDIRECTED GRAPHS / HERMITIAN MATRICES

General Results

Related Work for Products of Quadratic Forms

Generalizations for Products of Quadratic

Related Work for Powers of Quadratic Forms

Generalizations for Powers of Quadratic Forms

The Number of Walks and Degree Powers

Invalid Inequalities

Relaxed Inequalities Using Geometric Means

Restricted Graph Classes

Counterexamples for Special Cases

Semiregular Graphs

Trees

Subdivision Graphs

DIRECTED GRAPHS / NONSYMMETRIC MATRICES

Walks and Alternating Walks in Directed Graphs

Chebyshev’s Sum Inequality

Relaxed Inequalities Using Geometric Means

Alternating Matrix Powers and Alternating Walks

Powers of Row and Column Sums

Degree Powers and Walks in Directed Graphs

Row and Column Sums in Nonnegative Matrices

Other Inequalities for the Number of Walks

Geometric Means

APPLICATIONS

Bounds for the Largest Eigenvalue

Matrix Foundations

Graphs and Adjacency Matrices

Eigenvalue Moments and Energy

Iterated Kernels

Related Work

Our Results

Sidorenko’s Conjecture

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT002000
MATHEMATICS / Algebra / General
MAT036000
MATHEMATICS / Combinatorics