Tree sign pattern that permit eventual positivity

Abstract

Which sign patterns allow a matrix to be eventually positive (EP) is an open question that has received considerable attention. It is known that it is sufficient but not necessary that the positive part be primitive. Considered here are sign patterns, whose undirected graphs are trees (tree sign patterns, TSP's). For a few types of TSP's, it is known that the primitivity of the positive part is necessary and sufficient to allow EP. Here we show that this is so for all TSP's and conjecture that it is only so for trees. In addition, we consider the least power, at and beyond which all powers, of an EP matrix are positive, the “index”. For 2-by-2 matrices, the index is determined in terms of the entries. Unlike the case of primitive matrices, the index can be arbitrarily large.