A Combinatorial Approach to Matrix Theory and It's Applications

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.