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A Combinatorial Approach to Matrix Theory and Its Applications
Richard A. Brualdi and Dragoš Cvetković
Table of Contents
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