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 Konig 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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