Matrices with signed generalized inverses (Q1595118)

A complete characterization for \(m\times n\) matrices \(A\) with \(\rho(A)=n\) to have signed generalized inverses is given. The case \(\rho(A) <n \leq m\) is considered and complete characterizations are obtained for \(m \times n\) matrices with \(\rho(A)<n\leq m\) to have signed generalized inverses. 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. A characterization of matrices in a special triangular block form to have signed generalized inverses is also carried out.