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

Slobodan Filipovski

doi:10.26493/2590-9770.1280.4da

Slobodan Filipovski University of Primorska https://orcid.org/0000-0002-7286-4954

DOI: https://doi.org/10.26493/2590-9770.1280.4da

Keywords: Digraph, non-negative matrix, monotonicity, walk.

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.