1st Edition

A Combinatorial Approach to Matrix Theory and Its Applications

    283 Pages 44 B/W Illustrations
    by Chapman & Hall

    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.

    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.


    Richard A. Brualdi, Dragos Cvetkovic

    "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