A Combinatorial Approach to Matrix Theory and Its Applications: 1st Edition (Hardback) book cover

A Combinatorial Approach to Matrix Theory and Its Applications

1st Edition

By Richard A. Brualdi, Dragos Cvetkovic

Chapman and Hall/CRC

288 pages | 44 B/W Illus.

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pub: 2008-08-06
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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.


"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

Table of Contents




Some Classical Combinatorics


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


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


Electrical Engineering: Flow Graphs

Physics: Vibration of a Membrane

Chemistry: Unsaturated Hydrocarbons

Exercises appear at the end of each chapter.

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Combinatorics
SCIENCE / Physics