A fundamental matrix equation for finite sets

Let S={x1, x2, * * *, x,} be an n-set and let Sl, S2,. .. , Sm be subsets of S. Let A of size m by n be the incidence matrix for these subsets of S. We now regard x1, x2, .. , xn as independent indeterminates and define X=diag[x1, x2, *.. *, xnj. We then form the matrix product AXAT= Y, where AT denotes the transpose of the matrix A. The symmetric matrix Y has in its (i,j) position the sum of the indeterminates in S,r-Sj and consequently Ygives us a complete description of the intersection patterns S,(Si. The specialization x1=x2= ... =xn=1 of this basic matrix equation has been used extensively in the study of block designs. We give some other interesting applications of the matrix equation that involve subsets with various restricted intersection patterns. 1. The matrix equation. Let S= {x1, x2, * * , xj} be an n-set (a set of n elements) and let S,, S2,.. * , Sm be subsets of S. We set aij =1 if xi is a member of Si and we set aij=O if xj is not a member of Si. The resulting (0, 1)-matrix (1.1) A = [aij] of size m by n is the familiar incidence matrix for the subsets S,, S2, * , Sm of S. It is clear that A characterizes the configuration of subsets. Now let us regard xl, x2, * * *, xn, as independent indeterminates over the field of rational numbers and define (1.2) X= diag[xl, x2, ,xJ]. We then form the matrix product (1.3) AXAT Y The matrix AT denotes the transpose of the matrix A. The matrix Y is a symmetric matrix of order m. We know the structure of this matrix Received by the editors August 25, 1971. AMS 1970 subject classi{/cations. Primary 05B20, 05B30; Secondary 15A24.