An inequality for non-negative matrices II

Abstract

In a recent paper we proved that for an n× n matrix A with non-negative integer entries, there exist integers r, s with 0⩽ r< s⩽2 n such that A r ⩽ A s (cf. [Linear Algebra Appl. 290 (1999) 135]). Z. Bo [Austral. J. Combin. 21 (2000) 251] improved the bound 2 n to 3 n/2 . We give two results in this paper. First, we improve the bound to n+ g( n), where g is the Landau function. Thus we are close to the known lower bound of g( n) (cf. [Linear Algebra Appl. 290 (1999) 135]). Second, we show that if A is an irreducible matrix, then there is i=O( nlog n) such that A i ⩾ I. We also give examples where i=Ω(n logn/ log logn) .