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

Abstract

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