Matrices and set differences (Q790122)

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 \(X=diag[x_ 1,...,x_ t]\). 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.