Eigenvalue stability of Hermitian and normal matrices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915887008710656 |
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| author | Parusiński, Adam Rainer, Armin |
| author_facet | Parusiński, Adam Rainer, Armin |
| contents | The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian $d \times d$ matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces $W^{1,q}$, for all $1 \le q < \infty$, on bounded open domains. For $q=\infty$, the action is still well-defined and bounded but not continuous. We show that this stability result extends to normal matrices, where the eigenvalues are naturally interpreted as multivalued Sobolev functions in the sense of Almgren. Several applications are given, including the stability of singular values, condition numbers of matrices, surface area of eigenvalue graphs, and compact self-adjoint operators in Hilbert space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23056 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalue stability of Hermitian and normal matrices Parusiński, Adam Rainer, Armin Functional Analysis Analysis of PDEs Classical Analysis and ODEs Metric Geometry Spectral Theory The ordered eigenvalues define a Lipschitz map on the real vector space of Hermitian $d \times d$ matrices. We prove that this map acts continuously, but not uniformly continuously, by superposition on the Sobolev spaces $W^{1,q}$, for all $1 \le q < \infty$, on bounded open domains. For $q=\infty$, the action is still well-defined and bounded but not continuous. We show that this stability result extends to normal matrices, where the eigenvalues are naturally interpreted as multivalued Sobolev functions in the sense of Almgren. Several applications are given, including the stability of singular values, condition numbers of matrices, surface area of eigenvalue graphs, and compact self-adjoint operators in Hilbert space. |
| title | Eigenvalue stability of Hermitian and normal matrices |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Metric Geometry Spectral Theory |
| url | https://arxiv.org/abs/2603.23056 |