Consider an n \times n matrix polynomial P(\lambda). A spectral norm distance from P(\lambda) to the set of n n matrix polynomials that have a given scalar \mu \in C as a multiple eigenvalue was introduce and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of P(\lambda). In this paper, we extend this result to the case of two given distinct complex numbers \mu_1 and .\mu_2 First, we compute a lower bound for the spectral norm distance from P(\lambda) to the set of matrix polynomials that have \mu_1 and .\mu_2 as two eigenvalues. Then we construct an associated perturbation of P(\lambda) such that the perturbed matrix polynomial has two given scalars \mu_1 and \mu_2 in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of P(\lambda). Numerical examples are provided to illustrate the validity of the method.
