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 $\Sigma$ 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. 
E. Kokabifar, G.B. Loghmani, P.J. Psarrakos and S.M. Karbassi
math.NA