Summary: 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 = [$\alpha $ij ] requires eventual nonnegativity if and only if for every s, t such that $\alpha $st = - , the two principal submatrices of A indexed by $In(s)$ and $Out(t)$ require nilpotence. It is shown that Arequires eventual exponential positivity if and only if it requires exponential positivity, i.e., A is irreducible and its off-diagonal entries are nonnegative.

15A48, 05C50, 15A18

eventually nonnegative matrix, eventually positive matrix, eventually exponentially positive matrix, exponentially positive matrix, sign pattern, Perron-Frobenius