2025/09/30 by Vanni Noferini, Noferini, Vanni, Lauri Nyman +3
Computer Science · Mathematics · #math.NA #cs.NA
paper · pdf · doi:10.48550/arxiv.2509.26344
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, 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.