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

Abstract

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.