Sign k-potent sign patterns and ray k-potent ray patterns that admit k-potence

We prove that for each positive integer k, all irreducible k-potent ray pattern classes allow k-potence. We construct all irreducible real (complex) k-potent matrices in an irreducible sign (ray) k-potent sign (ray) pattern class. For a given reducible sign (ray) pattern that is sign (ray) k-potent for some positive integer k, we determine necessary conditions for that pattern to allow k-potence. This generalizes recent work by Lee and Park (2015) [2] who determined necessary conditions for a sign idempotent sign pattern to allow idempotence, and who showed that every irreducible, sign k-potent sign pattern must allow k-potence.