  A characterization of the structure of a regular matrix pencil obtained by a
bounded rank perturbation of another regular matrix pencil has been recently
obtained. The result generalizes the solution for the bounded rank perturbation
problem of a square constant matrix. When comparing the fixed rank perturbation
problem of a constant matrix with the bounded rank perturbation problem we
realize that both problems are of different nature; the first one is more
restrictive. In this paper we characterize the structure of a regular matrix
pencil obtained by a fixed rank perturbation of another regular matrix pencil.
We apply the result to find necessary and sufficient conditions for the
existence of a fixed rank perturbation such that the perturbed pencil has a
prescribed determinant. The results hold over fields with sufficient number of
elements.

Algebraic Geometry