Nearest matrix with multiple eigenvalues by Riemannian optimization
Vanni Noferini, Lauri Nyman, Federico Poloni
TL;DR
This work addresses the problem of finding the nearest perturbation $\Delta$ to a matrix $A\in\mathbb{C}^{n\times n}$ that yields a matrix with a multiple eigenvalue, i.e., the nearest matrix with multiple eigenvalues or, equivalently, the nearest defective matrix. It extends the Riemannian-optimization framework from the Riemann-Oracle approach to track both left and right eigenvectors on the complex Stiefel manifold $V_2(\mathbb{C}^n)$ and to impose arbitrary complex-linear constraints on $\Delta$ or $A+\Delta$, enabling structured perturbations like Toeplitz or sparse patterns. A variable-projection-based derivation provides closed-form expressions for the reduced objective, including a minimizer $\lambda_*$ and an explicit $f_{\varepsilon,y}(u,v)$, with scalable computation via Sherman–Morrison–Woodbury; the unstructured case yields simplified forms and rank-1 minimizers. Numerical experiments show competitive performance against state-of-the-art methods for unstructured perturbations and demonstrate the framework’s ability to handle structured perturbations, broadening applicability to eigenvalue-condition-number analysis and polynomial problems via companion matrices. The work offers practical algorithms and public code, enabling researchers to study stability under structured perturbations and to explore further extensions of Riemannian nearness problems in matrix theory.
Abstract
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.