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Matrices with signed generalized inverses. (English) Zbl 0967.15002
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.

##### MSC:
 15A09 Theory of matrix inversion and generalized inverses 15B48 Positive matrices and their generalizations; cones of matrices
##### Keywords:
signed generalized inverses; characterizations
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##### References:
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