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A Combinatorial Approach to Matrix Theory and Its Applications




ISBN 9781420082234
Published August 6, 2008 by Chapman and Hall/CRC
288 Pages - 44 B/W Illustrations

 
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Book Description

Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs 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 that are 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

Introduction
Graphs
Digraphs
Some Classical Combinatorics
Fields
Vector Spaces
Basic Matrix Operations
Basic Concepts
The König Digraph of a Matrix
Partitioned Matrices
Powers of Matrices
Matrix Powers and Digraphs
Circulant Matrices
Permutations with Restrictions
Determinants
Definition of the Determinant
Properties of Determinants
A Special Determinant Formula
Classical Definition of the Determinant
Laplace Development of the Determinant
Matrix Inverses
Adjoint and Its Determinant
Inverse of a Square Matrix
Graph-Theoretic Interpretation
Systems of Linear Equations
Solutions of Linear Systems
Cramer’s Formula
Solving Linear Systems by Digraphs
Signal Flow Digraphs of Linear Systems
Sparse Matrices
Spectrum of a Matrix
Eigenvectors and Eigenvalues
The Cayley–Hamilton Theorem
Similar Matrices and the JCF
Spectrum of Circulants
Nonnegative Matrices
Irreducible and Reducible Matrices
Primitive and Imprimitive Matrices
The Perron–Frobenius Theorem
Graph Spectra
Additional Topics
Tensor and Hadamard Product
Eigenvalue Inclusion Regions
Permanent and Sign-Nonsingular Matrices
Applications
Electrical Engineering: Flow Graphs
Physics: Vibration of a Membrane
Chemistry: Unsaturated Hydrocarbons
Exercises appear at the end of each chapter.

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Reviews

"The originality of the book lies – as its title indicates – in the use of combinatorial methods, specifically Graph Theory, in the treatment . . . An original and well-written textbook within whose pages even the most experienced reader should find something novel."

– Allan Solomon, Open University, in Contemporary Physics, May-June 2009, Vol. 50, No. 3