Graphs, Matrices, and Subspaces

Abstract

Graphs of the usual kind display functions. Graphs of a different kind (these have m edges connecting n nodes) lead to matrices. What y=f(x) is to calculus, matrices and subspaces are to linear algebra. Those are the central ideas, which students struggle with and to some extent master. This paper is about the incidence matrix A of a connected graph?and especially about its associated subspaces. For any m by n matrix there are four fundamental subspaces?two in Rn and two in Rm. They are the column spaces and nullspaces of A and AT. Their dimensions are related by the most important theorem in linear algebra. The second part of that theorem is the orthogonality of the subspaces. For the matrices we study, the dimensions and the orthogonality have a special meaning. Our goal is to show how examples from graphs illuminate the fundamental theorem of linear algebra. We start with the four subspaces (for any matrix). Then we state the theorem and construct a directed graph. For incidence matrices, the dimensions are easy to discover. But the subspaces themselves?this is where orthogonality helps. It is essential to connect the subspaces to the graph they come from. By specializing to incidence matrices, the laws of linear algebra become KirchhofFs laws. Every entry of an incidence matrix is 0 or 1 or -1. This continues to hold during elimination. All pivots and multipliers are ? 1. Therefore both factors in A =LU (lower triangular L and upper triangular U) also contain 0,1, -1. Remarkably this persists for the nullspaces of A and AT. All four subspaces have basis vectors with these exceptionally simple components. The matrices are not concocted for a textbook, they come from a model that is absolutely essential in pure and applied mathematics.