Sign pattern matrices and generalized inverses (Q1338510)

A sign pattern matrix is a matrix whose entries consist of element from the set \(\{+,-,0\}\). If \(A\) is a sign pattern matrix, then \(Q(A)\) denotes the set of all real \(n \times n\) matrices \(B\) such that the signs of the entries in \(B\) match the corresponding entries in \(A\). The class \({\mathcal G}\) of all square patterns \(A\) for which there exist \(B,C \in Q(A)\) where \(BCB=B\) is investigated. The main result is that, for nonnegative patterns, \({\mathcal G}\) coincides with the class of all square patterns \(A\) for which there exists a \(B \in Q(A)\) with \(B^ 3 = B\).