Journal or Book Title
Electronic Journal of Linear Algebra
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.
Ellison, Elisabeth M.; Hogben, Leslie; and Tsatsomeros, Michael J., "Sign patterns that require eventual positivity or require eventual nonnegativity" (2010). Mathematics Publications. 245.