On the distance from a matrix polynomial to matrix polynomials with some prescribed eigenvalues

Consider an n×n matrix polynomial P(λ) and a set Σ consisting of k≤n complex numbers. Recently, Kokabifar, Loghmani, Psarrakos and Karbassi studied a (weighted) spectral norm distance from P(λ) to the n×n matrix polynomials whose spectra contain the specified set Σ, under the assumption that all the entries of Σ are distinct. In this paper, the case in which some or all of the desired eigenvalues can be multiple is discussed. Lower and upper bounds for the distance are computed, and a perturbation of P(λ) associated to the upper bound is constructed. A detailed numerical example illustrates the efficiency and validity of the proposed computational method.