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

In this paper, we characterize sign k-potent sign pattern matrices that allow k-potence. In particular, the structure of a sign idempotent sign pattern matrix that allows idempotence is given. Thus an open problem posed by Eschenbach is affirmatively solved. We also extend these results to ray k-potent ray pattern matrices, providing the structure of a ray k-potent ray pattern matrix that allows k-potence.