Ela Sign Patterns That Require or Allow Power - Positivity

A matrix A is power-positive if some positive integer power of A is entrywise positive. A sign pattern A is shown to require power-positivity if and only if either A or −A is nonnegative and has a primitive digraph, or equivalently, either A or −A requires eventual positivity. A sign pattern A is shown to be potentially power-positive if and only if A or −A is potentially eventually positive. 1. Introduction. A matrix A ∈ R n×n is called power-positive [2, 10] if there is a positive integer k such that A k is entrywise positive (A k > 0). Note that if A is a power-positive matrix, then −A is also power-positive, because A k > 0 implies (−A) 2k > 0. If there is an odd positive integer k such that A k > 0, then A is called power-positive of odd exponent. Power-positive matrices have applications to the study of stability of competitive systems in economics; see, e.g., [7, 8, 9]. A real square matrix A is eventually positive if there exists a positive integer k 0 such that A k > 0 for all k ≥ k 0. An eventually positive matrix and its negative are both obviously power-positive.