Matrices with doubly signed generalized inverses (Q1855367)

This paper deals with doubly signed generalized inverses. A real matrix \(A\) is said to have a signed generalized inverse if \(B^\dagger\) and \(A^\dagger\) have the same sign pattern, for each matrix \(B\) with the same sign pattern as \(A\). By definition, a real matrix \(A\) is said to have doubly signed generalized inverse if both \(A\) and \(A^{\dagger}\) have signed generalized inverse, where \(A^{\dagger}\) is the Moore-Penrose inverse of \(A\). In this work a complete characterization of doubly signed generalized inverses is shown. The result presented here is a generalization of those well-known corresponding to signed generalized inverse and doubly \(S^2NS\) matrices.