A Combinatorial Approach to Matrix Theory and Its Applications
Chapman and Hall/CRC – 2008 – 288 pages
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.
"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
Some Classical Combinatorics
Basic Matrix Operations
The König Digraph of a Matrix
Powers of Matrices
Matrix Powers and Digraphs
Permutations with Restrictions
Definition of the Determinant
Properties of Determinants
A Special Determinant Formula
Classical Definition of the Determinant
Laplace Development of the Determinant
Adjoint and Its Determinant
Inverse of a Square Matrix
Systems of Linear Equations
Solutions of Linear Systems
Solving Linear Systems by Digraphs
Signal Flow Digraphs of Linear Systems
Spectrum of a Matrix
Eigenvectors and Eigenvalues
The Cayley–Hamilton Theorem
Similar Matrices and the JCF
Spectrum of Circulants
Irreducible and Reducible Matrices
Primitive and Imprimitive Matrices
The Perron–Frobenius Theorem
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.