An alternate proof of the monotonicity of the number of positive entries in nonnegative matrix powers

Abstract

Let A be a nonnegative real matrix of order n and f(A) denote the number of positive entries in A. In 2018, Xie proved that if f(A) ≤ 3 or f(A) ≥ n² − 2n + 2, then the sequence (f(A^k))_k = 1^∞ is monotone for positive integers k. In this note we give an alternate proof of this result by counting walks in a digraph of order n.