Products of Zero-One Matrices

Abstract

Let P be a finite set with p objects oj, j = 1, 2, … , p, and let {Si}, i = 1, 2, … , n, be a family of n subsets of P. The incidence matrix A = (aij) for the family {Si} is defined by the rules: aij = 1 if 0j, ∈ Si and aij = 0 if 0j ∉ Si. Then, if AAT = B = (bij) (where AT denotes the transpose of A), it is easy to see that bij = |Si ⋂ Sj|, i = 1, … , n, j = 1, … , n, so that the elements of B are integers with bii ⩾ bij ⩾ 0.