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

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