Fixing multiple eigenvalues by a minimal perturbation

Suppose we are given an n×n matrix, M, and a set of values, {λi}i=1m (m⩽n), and we wish to find the smallest perturbation in the 2-norm (i.e., spectral norm), ΔM, such that M-ΔM has the given eigenvalues λi. Some interesting results have been obtained for variants of this problem for fixing two distinct eigenvalues, fixing one double eigenvalue, and fixing a triple eigenvalue. This paper provides a geometric motivation for these results and also motivates their generalization. We also present numerical examples (both “successes” and “failures”) of fixing more than two eigenvalues by these generalizations.