Characterization of generalized inverses by a rank equation.
Consider the \(2 \times 2\) block-rank-equation rank\((A_{11},A_{12};A_{21},X) = \) rank\((A_{11})\) for blocks \(A_{11},A_{12}\), \(A_{21}\) and \(X\) of suitable sizes. If \(A_{11} \) is nonsingular then it is well-known that \(A_{11}^{-1}\) is the unique matrix \(X\) satisfying the above equation where \(A_{12}=A_{21}=I\). Similar results are known for the Moore-Penrose inverse and the group inverse of \(A_{11}\). In this paper, any outer inverse of \(A_{11} \in {\mathbb C}^{m \times n}\) is characterized by a similar rank equation (setting \(A_{12} = A_{11}X\) and \(A_{21}=XA_{11}\)). Moreover, the converse problem is solved for reflexive inverse and Drazin inverses. In fact, using the singular value (the core-nilpotent, resp.) decomposition, all matrices \(A_{12}\) and \(A_{21}\) such that a reflexive inverse (the Drazin inverse, resp.) of \(A_{11}\) is a solution of the above equation are found.
scientific article; zbMATH DE number 2062684