Nearest matrix with multiple eigenvalues by Riemannian optimization
This work provides a novel optimization framework for matrix nearness problems with eigenvalue constraints, benefiting researchers in numerical linear algebra and matrix theory.
The paper addresses the problem of finding the nearest matrix with multiple eigenvalues (or defective matrix) to a given square complex matrix. The proposed Riemannian optimization method achieves superior accuracy and efficiency compared to existing algorithms, with numerical experiments demonstrating improvements in convergence speed and solution quality.
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.