Nearest matrix with multiple eigenvalues by Riemannian optimization
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916005885771776 |
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| author | Noferini, Vanni Nyman, Lauri Poloni, Federico |
| author_facet | Noferini, Vanni Nyman, Lauri Poloni, Federico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearest matrix with multiple eigenvalues by Riemannian optimization Noferini, Vanni Nyman, Lauri Poloni, Federico Numerical Analysis 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. |
| title | Nearest matrix with multiple eigenvalues by Riemannian optimization |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.26344 |