A Combinatorial Approach to Matrix Theory and Its Applications
A Combinatorial Approach to Matrix Theory and Its Applications (Discrete Mathematics and Its Applications) Richard A. Brualdi, Dragos Cvetkovic,
Publisher: Chapman & Hall/CRC
ISBN: 142008223X
DDC: 512.9434
Edition: Hardcover; 2008-08-06
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
thatare 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.
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