Matrix Inequalities for Iterative Systems
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Matrix Inequalities for Iterative Systems
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Book 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