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 = [αij] requires eventual nonnegativity if and only if for every s, t such that αst = −, 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.
Comments
This article is published as Ellison, Elisabeth, Leslie Hogben, and Michael Tsatsomeros. "Sign patterns that require eventual positivity or require eventual nonnegativity." The Electronic Journal of Linear Algebra 19 (2010): 98-107. DOI: 10.13001/1081-3810.1350. Posted with permission.