# A Combinatorial Approach to Matrix Theory and Its Applications

Richard A. Brualdi, Dragos Cvetkovic

August 6, 2008 by Chapman and Hall/CRC
Reference - 288 Pages - 44 B/W Illustrations
ISBN 9781420082234 - CAT# C8223
Series: Discrete Mathematics and Its Applications

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### 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.