Which Nonnegative Matrices are Self-Inverse?

A binary relation, a digraph, and a graph are represented faithfully by their (square) adjacency matrices which are respectively: binary, binary with zero diagonal, and binary symmetric with zero diagonal. To determine the self-inverse matrices of these three types, we generalize the question to nonnegative matrices and then specialize the result. THEOREM. Let A be a nonnegative matrix with the property that A2 = In. Then there exists a permutation matrix P such that PAPT is a direct sum of 1-square matrices [1] and 2-square matrices of the formA=[0 1,a>O.