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

Consider an $n \times n$ matrix polynomial $P(\lambda)$. A spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was introduced 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 $\mu_1,\mu_2$ 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.