# Sign patterns that require eventual positivity or require eventual nonnegativity

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## Abstract

It is shown that a square sign pattern A requires eventual positivity if and only if it is nonnegative and primitive. Let the set of vertices in the digraph of A that have access to a vertex s be denoted by In(s) and the set of vertices to which t has access denoted by Out(t). It is shown that A = [α_{ij}] requires eventual nonnegativity if and only if for every s, t such that α_{st} = −, the two principal submatrices of A indexed by In(s) and Out(t) require nilpotence. It is shown that A requires eventual exponential positivity if and only if it requires exponential positivity, i.e., A is irreducible and its oﬀ-diagonal entries are nonnegative.

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