On the distance from a weakly normal matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

TL;DR

This paper refines existing bounds on the spectral norm distance from a matrix polynomial to those with a specified multiple eigenvalue, focusing on weakly normal polynomials and demonstrating an effective modification method.

Contribution

It provides a refined upper bound for weakly normal matrix polynomials and introduces a modification method with verified efficiency.

Findings

Refined upper bound for weakly normal matrix polynomials.

Implementation of a modification method.

Efficiency verified through an illustrative example.

Abstract

Consider an $n \times n$ matrix polynomial $P(\lambda)$. An upper bound for a spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was recently obtained by Papathanasiou and Psarrakos (2008). This paper concerns a refinement of this result for the case of weakly normal matrix polynomials. A modification method is implemented and its efficiency is verified by an illustrative example.