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

Consider an $n\times n$ matrix polynomial $P(\lambda )$ and a set $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$, 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.