1st Edition

# A Combinatorial Approach to Matrix Theory and Its Applications

**Also available as eBook on:**

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.*

### Biography

Richard A. Brualdi, Dragos Cvetkovic

"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