Finite Rank Perturbations of Linear Relations and Matrix Pencils

Abstract

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by $$n+1$$ n + 1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.

The spectrum and the Weyr characteristics of operator pencils and linear relations

“…In this section we investigate one-dimensional perturbations of regular operator pencils based on the underlying kernel and range representations. We derive formulas for the change of the Weyr characteristic from a combination of Theorem 4.5 and a recent perturbation result for linear relations [31]. The following notion of perturbations for linear relations was used in [4,31].…”

“…We derive formulas for the change of the Weyr characteristic from a combination of Theorem 4.5 and a recent perturbation result for linear relations [31]. The following notion of perturbations for linear relations was used in [4,31].…”

“…The proof of (4) in finite dimensions are based on invariant factors and their perturbations and, hence, cannot be applied here. We follow a different approach which is based on a recent perturbation result on the Weyr characteristic of linear relations from [31]. Linear relations are just subspaces of X × X, which allow a rich spectral theory, see e.g.…”

On Jordan chains of perturbed matrix pencils

“…Recently, there have been obtained bounds for the changes of the generalized Weyr characteristic of a complex matrix pencil (see Remark 3.3 below) perturbed by another pencil of the form w(su * − v * ) ( [11,Theorem 7.8]). This has been done relating the Jordan chains of a square pencil with those of a linear relation.…”

“…We have extended the notion of Jordan chain to possibly non square matrix pencils and have expressed the generalized Weyr characteristic of a pencil in terms of its Kronecker structure. This has already been done in [11] for square pencils.…”