Let A be a matrix of size m by n with elements in a field F. Let X = diag[ x 1,…, x n ] be a diagonal matrix of order n, where x 1,…, x n are n independent indeterminates over F. Throughout the paper we investigate the matrix equation AXA T = Y, where A T denotes the transpose of the matrix A. We call this equation the fundamental matrix equation of set intersections and we call the symmetric matrix Y of order m the set intersection matrix defined by A. The terminology arises from the important special case in which A is a (0, 1)-matrix. Then A may be regarded as the incidence matrix for m subsets of an n-set and in this special case the equation gives us a complete description of the pairwise intersection patterns of the subsets. Moreover, it displays this information in an exceedingly compact form. Our paper is primarily concerned with the derivation of four theorems involving the set intersection matrix Y. Two of the theorems reveal the extent to which the polynomial det( Y) and the characteristic polynomial f( z) of Y characterize the set intersection matrix Y. The remaining two theorems determine simple necessary and sufficient conditions for the irreducibility of the polynomial det( Y) in the polynomial ring F ∗ = F[x 1,…,x n] and for the irreducibility of the characteristic polynomial f( z) of Y in the polynomial ring F ∗[z] . Our results are of interest from both a matrix theoretic and combinatorial point of view.