SIGN PATTERN MATRICES AND GENERALIZED INVERSES

If A is an n x n sign pattern matrix, then Q(A) denotes the set of all real n x n matrices B such that the signs of the entries in B match the corresponding entries in A. In this paper, we consider various requires/allows questions connected with sign pattern matrices and generalized inverses. In particular, we investigate the class G of all square patterns A for which there exist B, C is-an-element-of Q(A) where BCB = B. For nonnegative patterns, we characterize G and show that G coincides with the class of all square patterns A for which there exists B is-an-element-of Q(A) where B3 = B. Nonnegative square patterns that allow an idempotent and those that allow a (1,3)-inverse are each characterized. Some interesting open questions are also indicated.