Matrices with signed generalized inverses

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. In this paper, we give complete characterizations of the m x n matrices A which have signed generalized inverses, for both cases rhop (A) = n and rho (A) < n (where rho (A) is the term rank of A, and without loss of generality we assume n less than or equal to m), and thus solve a problem proposed in [B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056]. Using these characterizations, we are also able to show that the property of having a signed generalized inverse for a matrix A is inherited by all the submatrices B of A with rho (B)= rho (A) and is also inherited by all those matrices A(1) with rho (A(1)) = rho (A) which can be obtained from A by replacing some nonzero entries of A by zero. We also consider several special cases of a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, MA, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterization of the matrices in a special triangular block form to have signed generalized inverses. (C) 2001 Elsevier Science Inc. All rights reserved.