A matrix sequence [formula omitted] might converge even if the matrix A is not primitive

Abstract

It is well-known that, for an irreducible Boolean (0,1)-matrix A, the matrix sequence {Am}m=1∞ 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 {Γ(Am)}m=1∞ 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 i≠j, 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 {Γ(Am)}m=1∞ for a Boolean (0,1)-matrix A whose digraph has at most two strong components. We show that {Γ(Am)}m=1∞ 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 {Γ(Am)}m=1∞ converges, and find the limit of {Γ(Am)}m=1∞ in terms of its digraph when it converges. We derive these results in terms of the competition graph of the digraph of A.