Uniqueness of size-2 positive semidefinite matrix factorizations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914987301142528 |
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| author | Dawson, Kristen Hoşten, Serkan Kubjas, Kaie Metsälampi, Lilja |
| author_facet | Dawson, Kristen Hoşten, Serkan Kubjas, Kaie Metsälampi, Lilja |
| contents | We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of size-2 positive semidefinite matrix factorizations Dawson, Kristen Hoşten, Serkan Kubjas, Kaie Metsälampi, Lilja Metric Geometry Optimization and Control 15B48, 15A23, 14P10, 52B05 We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions. |
| title | Uniqueness of size-2 positive semidefinite matrix factorizations |
| topic | Metric Geometry Optimization and Control 15B48, 15A23, 14P10, 52B05 |
| url | https://arxiv.org/abs/2410.18891 |