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

The 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, losing 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 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 after perturbation. The resulting bounds are expressed in terms of the corresponding component of the unperturbed pencil. To verify that the bounds are reachable, we characterize the Weyr characteristic of a pencil obtained from another one by removing or adding one row. The results hold for algebraically closed fields.