Matrices and set differences

Abstract

Let A and B be (0, 1)-matrices of sizes m by t and t by n, respectively. Let x 1, …, x t denote t independent indeterminates over the rational field Q and define X = diag[ x t , …, x t ]. We study the matrix equation AXB = Y. We first discuss its combinatorial significance relative to topics such as set intersections and the Marica-Schönheim theorem on set differences. We then prove the following theorem concerning the matrix Y. Suppose that the matrix Y of size m by n has rank m. Then Y contains m distinct nonzero elements, one in each of the m rows of Y.