The solution of a problem on matrices having 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. In this paper, we solve a problem proposed in [R.A. Brualdi, B.L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, Cambridge, 1995; B.L. Shader, SIAM. J. Matrix Anal. Appl. 16 (1995) 1056] about the characterizations of the matrices with a special lower triangular blocked form to have a signed generalized inverse. Using this characterization and the fact that every matrix which has a signed generalized inverse and full column term rank and no zero rows is permutation equivalent to a matrix of this form, we give two algorithms for determining whether or not a matrix with full column term rank, or a general matrix, has a signed generalized inverse.