A matrix sequence might converge even if the matrix A is not primitive

It is well-known that, for an irreducible Boolean (0, 1)-matrix A, the matrix sequence converges if and only if A is primitive. In this paper, we introduce an operation Γ on the set of Boolean (0, 1)-matrices such that a matrix sequence might converge even if the matrix A is not primitive. Given a Boolean (0, 1)-matrix A, we define a matrix Γ(A) so that the (i, j)-entry of Γ(A) equals 0 if for , the inner product of the ith row and jth row of A is 0 and equals 1 otherwise.The aim of this paper is to study the convergence of for a Boolean (0, 1)-matrix A whose digraph has at most two strong components. We show that converges to a very special type of matrix as m increases if A is an irreducible Boolean matrix. Furthermore, we completely characterize a Boolean (0, 1)-matrix A whose digraph has exactly two strongly connected components and for which converges, and find the limit of in terms of its digraph when it converges. We derive these results in terms of the competition graph of the digraph of A.