Sign-patterns which require a positive eigenvalue

Abstract

We investigate matrices which have a positive eigenvalue by virtue of their sign-pattern and regardless of the magnitudes of the entries. When all the off-diagonal entries are nonzero, we show that an n × n sign-pattern, n = 3, 4, requires a positive eigenvalue if and only if it has at least one nonnegative diagonal entry and every cycle of length greater than one in its signed digraph is positive. When n = 3, 4, or when not all off-diagonal entries are nonzero, positivity of the cycles of length greater than one is no longer necessary. In the course of proving these results we observe certain necessary and certain sufficient *

Sign patterns that require eventual exponential nonnegativity

“…Sign patterns that require or allow various special eigenstructures have been studied (see, e.g., [38,16,5,44,14,11,12,15]). The allows problem for the properties eventual nonnegativity and eventual exponential nonnegativity are wide open, whereas those sign patterns that require eventual positivity, eventual nonnegativity, and eventual exponential positivity have been completely characterized [19] and Chapter 2 of this dissertation presents results on sign patterns that require eventual exponential nonnegativity.…”