A Combinatorial Approach to Matrix Theory and Its Applications
This book is about expressing and proving basic facts of linear algebra (including some more advanced topics) in the language of oriented (directed) graphs (digraphs). The graphs are weighted, with the edges of zero weight regarded as removed from the base set of all edges. The weights come from elements of matrices. With a matrix A = [aij], one can always associate a bipartite graph in which the edges go from symbols representing rows to symbols representing columns so that the edge from the row i to the column j gets the weight aij. This representation allows an interpretation of matrix products but for further purposes one has to consider the digraph in which the ith row and ith column are represented by the same symbol. Of course, this presupposes that the matrix A is square. By representing permutations as subgraphs one can define determinants as a sum of weighted paths, where the weight of a path is by default the product of the weights. The standard proofs of determinant properties, Laplace development, inverses and Cramer’s formula are then formulated in this setting.
The main idea seems to be to make the basics of linear algebra more understandable and concrete. There are many exercises and the intended audience obviously includes electrical engineers and students of electrical engineering. The basics comprise a bit more than half of the book. They are followed by an interesting proof of the Cayley-Hamilton theorem in this language and also by a proof of the Jordan Canonical Form. The Perron-Frobenius theorem also appears but without proof. Digraphs are used to discuss some of its consequences and some related notions. The last chapter presents applications from three areas. One is concerned with flow graphs (electrical engineering), another with vibrations of membranes (physics) and the last with unsaturated hydrocarbons (chemistry). The book is written very carefully and the speed of exposition is leisurely, in particular at the beginning. The book can serve well as a supplementary material that sheds light upon subjects that are sometimes regarded as too abstract. It can help the reader to grasp such subjects with more confidence and understanding.