A theorem in combinatorial matrix theory

Let Z be a matrix of order n, and suppose that the elements of Z consist of only two elements x and y, which are elements of a field F. We call Z an (x,y)-matrix over F. In this paper we study the matrix equation ZEZ^T = D+λJ, where Z is a nonsingular (x,y)-matrix over F, Z^T is the transpose of Z, D and E are nonsingular diagonal matrices, J is the matrix of 1's and λ is an element of F. Our main theorem shows that the column sums of Z are severely restricted. This result generalizes a number of earlier investigations that deal with symmetric block designs and related configurations. The problems that emerge are of interest from both a combinatorial and a matrix theoretic point of view.