Sign Idempotent Matrices and Generalized Inverses

A matrix whose entries consist of $\{+,-,0\}$ is called a sign pattern matrix. Let $Q(A)$ denote the set of all real $n\times n$ matrices $B$ such that the signs of entries in $B$ match the corresponding entries in $A$. For nonnegative sign patterns, sign idempotent patterns have been characterized. In this paper, we firstly give an equivalent proposition to characterize general sign idempotent matrices (sign idempotent). Next, we study properties of a class of matrices which can be generalized permutationally similar to specialized sign patterns. Finally, we consider the relationships among the allowance of idempotent, generalized inverses and the allowance of tripotent in symmetric sign idempotent patterns.