On the class of \(\{1,2,4\}\) inverses as solution of a linear problem (Q1275305)

Let \(A\) be an \(m\times n\) real matrix. The paper works out a characterization of those \(n\times m\) matrices \(X\) that satisfy the three conditions (1) \(AXA=A\); (2) \(XAX=X\); and (4) \((XA)^T= XA\). By ignoring condition (3) (i.e., \((AX)^T= AX\)), which in the presence of the other three conditions would force \(X\) to be the usual (and unique!) Moore-Penrose generalized inverse \(A^+\) of \(A\), the author is able to show that the matrices \(X\) are in bijective correspondence with the set of vector space complements of the column space of \(A\).