Computing rank-deficiency of rectangular matrix pencils

Rectangular matrix pencils arise in many contexts in Linear Control Theory, for example the PBH Test for Controllability and the computation of transmission zeros. In this paper, we examine a method to find the distance from an arbitrary given pencil to a nearby rank-deficient one, in light of the fact that rectangular pencils are generically full-rank. We propose an experimental computational method that exhibits quadratic convergence to a local minimum of this distance function. This partially answers the question of the existence of a rank-deficient pencil in a neighborhood of a given pencil. We use the Algorithm to illustrate some limitations of previous algorithms to measure this distance.