One dimensional perturbation problem for linear relations

Abstract

Given two linear relations S and T in C n, we characterize when there exist linear relations S˜ and T˜ in C n, strictly equivalent to S and T, respectively, such that S˜ and T˜ are one dimensional perturbations of each other; i.e., S˜ ∩ T˜ is a linear subspace of codimension at most one both in S˜ and in T˜, but it cannot be equal to both of them simultaneously. The result is achieved relating one dimensional perturbations of linear relations with rank-one perturbations of matrix pencils.