Gram-like matrix preserving extensions and completions of noncommutative polynomials
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915360585809920 |
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| author | Mukherjee, Arijit Sutradhar, Arindam |
| author_facet | Mukherjee, Arijit Sutradhar, Arindam |
| contents | Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a noncommutative polynomial to a SOHS keeping the Gram matrices unperturbed. As a remedy, we introduce an equally significant notion of Gram-like matrices and provide linear algebraic techniques to get the desired extensions. We further use positive semidefinite completion problem to get SOHS and provide criteria in terms of chordal graphs and 2-regular projective algebraic sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12063 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gram-like matrix preserving extensions and completions of noncommutative polynomials Mukherjee, Arijit Sutradhar, Arindam Optimization and Control Commutative Algebra Algebraic Geometry Functional Analysis 14P05, 14N25, 13D02, 05C50, 11E25, 90C90 Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a noncommutative polynomial to a SOHS keeping the Gram matrices unperturbed. As a remedy, we introduce an equally significant notion of Gram-like matrices and provide linear algebraic techniques to get the desired extensions. We further use positive semidefinite completion problem to get SOHS and provide criteria in terms of chordal graphs and 2-regular projective algebraic sets. |
| title | Gram-like matrix preserving extensions and completions of noncommutative polynomials |
| topic | Optimization and Control Commutative Algebra Algebraic Geometry Functional Analysis 14P05, 14N25, 13D02, 05C50, 11E25, 90C90 |
| url | https://arxiv.org/abs/2501.12063 |