Structures preserved by generalized inversion and Schur complementation

In this paper we investigate the inheritance of certain structures under generalized matrix inversion. These structures contain the case of rank structures, and the case of displacement structures. We do this in an intertwined way, in the sense that we develop an argument that can be used for deriving the results for displacement structures from thoses for rank structures. We pay particular attention to the Moore-Penrose generalized inverse, showing that for the cases of most interest, the ranks of the structure satisfied by the Moore-Penrose inverse can at most double with respect to the original ranks. We consider also the case of inheritance of structure by generalized Schur complements.