Sign-patterns which require a positive eigenvalue
Steve Kirkland, Judith J. McDonald, Michael J. Tsatsomeros · Linear and Multilinear Algebra · 1996
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× nsign-patternn≠ 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 conditions for a general sign-pattern to require a positive eigenvalue. We also identify and construct more classes of sign-patterns which require a positive eigenvalue.