Sign idempotent sign pattern matrices that allow idempotence.

It is well known that not all sign idempotent sign pattern matrices allow idempotence. In this paper, we found a necessary and sufficient condition for a sign idempotent sign pattern matrix to allow idempotence. In addition, we investigated the nonnegativity of sign idempotent sign pattern matrices that allow idempotence. Finally, we show that if A is a sign idempotent sign pattern matrix, then A allows idempotence if and only if | A | = [ | a i j | ] also allows idempotence. [ABSTRACT FROM AUTHOR]