A Combinatorial Approach to Matrix Theory and Its Applications (Hardcover)

Introduction Graphs Digraphs Some Classical Combinatorics Fields Vector Spaces Basic Matrix Operations Basic Concepts The Konig 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.