Matrices with doubly signed generalized inverses Original Research Article

A real matrix A is said to have a signed generalized inverse, if the sign pattern of its generalized inverse A+ is uniquely determined by the sign pattern of A. A is said to have doubly signed generalized inverse, if both A and A+ have a signed generalized inverse. Matrices with doubly signed generalized inverses are generalizations of those matrices A such that both A and A−1 are S2NS matrices which are studied in [Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, 1995; Linear Algebra Appl. 232 (1996) 97]. In this paper we give a complete characterization of matrices with doubly signed generalized inverses. We show that if A is a real matrix with no zero rows and no zero columns, then A has doubly signed generalized inverses if and only if A is permutation equivalent to a direct sum of matrices of the following types: a row (or a column) with no zero entries; fully indecomposable S2NS matrices of order 1 or 2; matrices of the blocked form image where D1 and D2 are both invertible diagonal matrices.