Nearest matrix with multiple eigenvalues by Riemannian optimization

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix.

To this goal, we extend the general framework described in [M.

Gnazzo, V.

Noferini, L.

Nyman, F.

Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found.

Comput.

Math., To appear] and based on variable projection and Riemannian optimization, allowing the ambient manifold to simultaneously track left and right eigenvectors.

Our method also allows us to impose arbitrary complex-linear constraints on either the perturbation or the perturbed matrix; this can be useful to study structured eigenvalue condition numbers.

We present numerical experiments, comparing with preexisting algorithms.