A Combinatorial Approach to Matrix Theory and its Applications CRC - Brualdi, Cvetkovic 2009

Through combinatorial & graph theoretic tools, this self-contained reference helps readers understand the fundamentals of matrix theory & its applications to science. It develops the theory using graphs to explain the basic matrix construction, formulas, computations, ideas, & results.

Features

Places combinatorial and graph-theoretical tools at the forefront of the development of matrix theory

Fosters a better understanding of matrix theory by using graphs to explain basic matrix construction, formulas, computations, ideas, and results

Presents material rarely found in other books at this level, including Gersgorin’s theorem and its extensions, the Kronecker product of matrices, sign-nonsingular matrices, and the evaluation of the permanent matrix

Includes a combinatorial argument for the classical Cayley–Hamilton theorem and a combinatorial proof of the Jordan canonical form of a matrix

Describes several applications of matrices in electrical engineering, physics, and chemistry

Summary

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.

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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