On the distance from a matrix polynomial to matrix polynomials with $k$ prescribed distinct eigenvalues

TL;DR

This paper investigates the spectral norm distance from a matrix polynomial to those with a prescribed set of eigenvalues, providing bounds and optimal perturbations, supported by numerical examples.

Contribution

It introduces bounds for the distance to matrix polynomials with specified eigenvalues and constructs an optimal perturbation, advancing spectral analysis methods.

Findings

Derived upper and lower bounds for the distance.

Constructed an optimal perturbation related to the upper bound.

Numerical examples demonstrate the bounds' effectiveness.

Abstract

Consider an $n\times n$ matrix polynomial $P(\lambda)$ and a set $\Sigma$ consisting of $k \le n$ distinct complex numbers. In this paper, a (weighted) spectral norm distance from $P(\lambda)$ to the matrix polynomials whose spectra include the specified set $\Sigma$, is defined and studied. An upper and a lower bounds for this distance are obtained, and an optimal perturbation of $P(\lambda)$ associated to the upper bound is constructed. Numerical examples are given to illustrate the efficiency of the proposed bounds.