Taubig Matrix Inequalities for Iterative Systems

Erscheinungsjahr 2016
ISBN: 978-1-4987-7779-7
Verlag: CRC Press
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 218 Seiten

This book investigates inequalities for weighted entry sums of matrix powers with applications ranging from mathematics and computer science to biology, physics, and chemistry. Unifying the research for products and powers of sesquilinear forms derived from powers of Hermitian, positive-semi definite, and nonnegative matrices, the results shows that certain inequalities are not valid in general but only in special cases. Furthermore, the book demonstrate how to translate the Hermitian matrix results into results for alternating powers of general rectangular matrices and refines inequalities for nonnegative matrices.

Autoren/Hrsg.

Taubig, Hanjo

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

INTRODUCTION. Notation and Basic Facts. Matrices and Vectors. Graphs. Number of Walks in Graphs. Entry Sums and Matrix Powers. Subsets, Sub-matrices, 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. UNDIRECTED GRAPHS, SYMMETRIC AND HERMITIAN MATRICES. General Results. Related Work for Products of Quadratic Forms. Generalizations for Products of Quadratic Forms. Related Work for Powers of Quadratic Forms. Generalizations for Powers of Quadratic Forms. The Number of Walks and Degree Powers. Invalid Inequalities. Relaxed Inequalities. Restricted Graph Classes. Counterexamples for Special Cases. Trees. Subdivision Graphs. DIRECTED GRAPHS AND 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.