The factors of a design matrix

Abstract

AbstractLet X and Y be integral matrices of order n > 1 and suppose that these matrices satisfy the matrix equation XY = B, where B is a matrix with k in the n main diagonal positions and λ and μ in all other positions. Suppose further that k, λ, and μ are nonnegative integers and that λ occurs exactly the same number of times in each line of B and that a similar situation holds for μ. We call X and Y the factors of the design matrix B. The matrix equation described above embraces a vast category of combinatorial configurations that are characterized by square incidence matrices. We investigate the factors of design matrices and prove a duality theorem for the factors of certain “quadratic” design matrices. This result may be regarded as a strong generalization of Connor's duality theorem on symmetric group divisible designs. We conclude with a brief discussion of certain special factors of design matrices that are of particular interest to us