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Consider an $n \times n$ matrix polynomial $P(\lambda)$. A spectral normdistance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials thathave a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was introducedand obtained by Papathanasiou and Psarrakos. They computed lower and upperbounds for this distance, constructing an associated perturbation of$P(\lambda)$. In this paper, we extend this result to the case of two givendistinct complex numbers $\mu_{1}$ and $\mu_{2}$. First, we compute a lowerbound for the spectral norm distance from $P(\lambda)$ to the set of matrixpolynomials that have $\mu_1,\mu_2$ as two eigenvalues. Then we construct anassociated perturbation of $P(\lambda)$, such that the perturbed matrixpolynomial 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 themethod.

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