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

Consider an n × n matrix polynomial p(λ). a spectral norm distance from p(λ) to the set of n × n matrix polynomials that havea given scalar µ ∈ c as a multiple eigenvalue was introducedand obtained by papathanasiou and psarrakos. they computedlower and upper bounds for this distance, constructing an associated perturbation of p(λ). in this paper, we extend this resultto the case of two given distinct complex numbers µ1 and µ2.first, we compute a lower bound for the spectral norm distancefrom p(λ) to the set of matrix polynomials that have µ1, µ2 astwo eigenvalues. then we construct an associated perturbationof p(λ) such that the perturbed matrix polynomial has two givenscalars µ1 and µ2 in its spectrum. finally, we derive an upperbound for the distance by the constructed perturbation of p(λ).numerical examples are provided to illustrate the validity of themethod.