When does the Moore-Penrose inverse flip? - Hartwig, Robert E. (2012)
In this paper, we give necessary and sufficient conditions for the matrix $\mxl{cc}a&0\\b&d\mxr$, over a *-regular ring, to have a Moore-Penrose inverse of four different types, corresponding to the four cases where the zero element can stand. In particular, we study the case where the Moore-Penrose inverse of the matrix flips.