invariants of the associated characteristic pencil.

In the last decades rank perturbations of matrix pencils have been analyzed in many papers from different approaches. The problem has been studied generically, i.e., when the perturbation $P(s)$ belongs to an open and dense subset of the set of pencils of rank less than or equal to $r$, for a given integer $r$. In other cases, the pencil $P(s)$ is an arbitrary perturbation belonging to the whole set of pencils of rank less than or equal to $r$. In this paper we follow the second approach.

From a generic point of view, changes in the Jordan structure of a square constant matrix or in the Weierstrass structure of a regular pencil corresponding to a fixed eigenvalue after low rank perturbations have been studied, among others, in [@Batzke14; @BaMeRaRo16; @TeDo16; @TeDoMo08; @MeMeRaRo11; @MoDo03; @Sa02; @Sa04]. See also the references therein.

The case where the perturbation is an arbitrary square matrix $P$ or a regular matrix pencil $P(s)$ has also been studied by several authors. For square constant matrices and a constant perturbation of bounded rank $r$, a solution is given in [@Silva88_1] and [@Za91]. For $r=1$ the problem was already solved in [@Th80]. The 