Structures preserved by matrix inversion

In this paper we investigate some matrix structures on $\cee^{n\times n}$ 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 $\Lam_k\in\mathbb{C}^{m \times m}$, for suitable m, and with Fiedler and Markham's theorem corresponding to the limiting cases $\Lam_k\to 0$ and $\Lam_k\to\infty I$.