invariants of a pair of matrices, i.e., the Kronecker invariants of the associated characteristic pencil, under a constant perturbation of bounded rank is solved in [@DoSt14]. Here, the authors find the solution relating the problem to a matrix pencil completion problem.

In this paper we study arbitrary rank-one perturbations of matrix pencils. We solve the problem transforming it into a matrix pencil completion problem. The solution obtained holds for arbitrary fields.

The paper is organized as follows. In Section \[secpreliminaries\] we introduce the notation, basic definitions and preliminary results. In Section \[secproblem\] we establish the problem which we are going to study and relate it to a matrix pencil completion problem. Then, in Section \[seccompletion\] we introduce previous results about completion of matrix pencils which will be needed later. In Section \[sectechnical\] some thecnical lemmas are proved. In Section \[secmain\], a solution to the stated rank-one perturbation problem is given in Theorem \[maintheogen\]. Finally, in Section \[secconclusions\] we summarize the main contributions of the paper.

Preliminaries {#secpreliminaries}
=============

Let ${\mathbb F}$ be a field. ${\mathbb F}[s]$ denotes the ring of polynomials in the indeterminate $s$ with coefficients in ${\mathbb F}$ and ${\mathbb F}[s, t]$ the ring of polynomials in two 