A Combinatorial Approach to Matrix Theory and Its Applications

by Brualdi; Richard A.

• ISBN13: 9781420082234
• ISBN10: 142008223X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-08-06
• Publisher: Chapman & Hall/

Summary

Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applicationsemploys combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices.After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices thatare part of the Perron'Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry.Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.

Table of Contents

Preface	p. xi
Dedication	p. xv
Introduction	p. 1
Graphs	p. 2
Digraphs	p. 8
Some Classical Combinatorics	p. 10
Fields	p. 13
Vector Spaces	p. 17
Exercises	p. 23
Basic Matrix Operations	p. 27
Basic Concepts	p. 27
The Konig Digraph of a Matrix	p. 35
Partitioned Matrices	p. 43
Exercises	p. 46
Powers of Matrices	p. 49
Matrix Powers and Digraphs	p. 49
Circulant Matrices	p. 58
Permutations with Restrictions	p. 59
Exercises	p. 60
Determinants	p. 63
Definition of the Determinant	p. 63
Properties of Determinants	p. 72
A Special Determinant Formula	p. 85
Classical Definition of the Determinant	p. 87
Laplace Determinant Development	p. 91
Exercises	p. 94
Matrix Inverses	p. 97
Adjoint and Its Determinant	p. 97
Inverse of a Square Matrix	p. 101
Graph-Theoretic Interpretation	p. 103
Exercises	p. 107
Systems of Linear Equations	p. 109
Solutions of Linear Systems	p. 109
Cramer's Formula	p. 118
Solving Linear Systems by Digraphs	p. 121
Signal Flow Digraphs of Linear Systems	p. 127
Sparse Matrices	p. 133
Exercises	p. 137
Spectrum of a Matrix	p. 139
Eigenvectors and Eigenvalues	p. 139
The Cayley-Hamilton Theorem	p. 147
Similar Matrices and the JCF	p. 150
Spectrum of Circulants	p. 165
Exercises	p. 167
Nonnegative Matrices	p. 171
Irreducible and Reducible Matrices	p. 171
Primitive and Imprimitive Matrices	p. 174
The Perron-Frobenius Theorem	p. 179
Graph Spectra	p. 184
Exercises	p. 188
Additional Topics	p. 191
Tensor and Hadamard Product	p. 191
Eigenvalue Inclusion Regions	p. 196
Permanent and SNS-Matrices	p. 204
Exercises	p. 215
Applications	p. 217
Electrical Engineering: Flow Graphs	p. 218
Physics: Vibration of a Membrane	p. 224
Chemistry: Unsaturated Hydrocarbons	p. 229
Exercises	p. 240
Coda	p. 241
Answers and Hints	p. 245
Bibliography	p. 253
Index	p. 261
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