Bounds for the change of the Weyr characteristic of matrix pencils after 1-rank perturbations

TL;DR

This paper introduces bounds on how the Weyr characteristic of a matrix pencil changes under rank-one perturbations, simplifying the understanding of Kronecker structure modifications.

Contribution

It defines the Weyr characteristic for matrix pencils and provides bounds on its change under rank-one perturbations, clarifying complex existing conditions.

Findings

Bounds are expressed solely in terms of original Weyr components.

Characterization of partitions after adding or removing a row.

Results applicable over algebraically closed fields.

Abstract

The complete characterization of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one is known, and it is stated in terms of very involved conditions. This paper is devoted to, loosing accuracy, better understand the meaning of those conditions. The Kronecker structure of a pencil is determined by the sequences of the column and row minimal indices and of the partial multiplicities of the eigenvalues. We introduce the Weyr characteristic of a matrix pencil as the collection of the conjugate partitions of the previous sequences and provide bounds for the change of each one of the Weyr partitions when the pencil is perturbed by a pencil of rank one. For each one of the Weyr components, the resulting bounds are expressed only in terms of the corresponding component of the unperturbed pencil. In order to verify that the bounds are reachable, we also characterize the partitions of the Weyr characteristic of a pencil obtained from another one by removing or adding one row. The results hold for algebraically closed fields