Fixing two eigenvalues by a minimal perturbation

The optimal perturbation, ΔM, to a matrix, M, such that M − ΔM has a given eigenvalue λ0 is given by the Eckart-Young theorem. This perturbation is optimal in the sense that ∥ΔM∥2 is minimal. In this article, we present a generalization, finding ∥ΔM∥2 optimal perturbations of M such that M − ΔM has two of given eigenvalues. This result also generalizes recent work by Malyshev.