Structures preserved by matrix inversion

In this paper we investigate some matrix structures on C-nXn that have a good behavior under matrix inversion. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, the inverse matrix also must have a low rank submatrix, which we can explicitly determine. This allows us to generalize a theorem due to Fiedler and Markham. The generalization consists in the fact that our rank structures may have a certain correction term, which we call the shift matrix Lambda(k) is an element of C-mXm, for suitable m, and with Fiedler and Markham's theorem corresponding to the limiting cases Lambda(k) -> 0 and Lambda(k) -> infinity I.