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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.02314 |
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| _version_ | 1866914527664144384 |
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| author | Garbe, Frederik Wei, Fan |
| author_facet | Garbe, Frederik Wei, Fan |
| contents | A well-known fact in linear algebra is that $A^T A$ is always positive semi-definite for any real matrix $A$. We consider a generalization of this fact via the following decision problem. Given a symbolic product of length $k$, consisting of $\ell$ variables and their transposes, such as $ABB^TCA^T$, does there exist an $n\in\mathbb N$ and an assignment of matrices from $\mathbb R^{n\times n}$ such that the resulting matrix product has a negative eigenvalue? We show that this problem is decidable and provide a simple characterization of those symbolic products that have only non-negative real eigenvalues for any assignment of matrices. This characterization can also be understood as a matrix analogue of the positive graph conjecture by Camarena, Csóka, Hubai, Lippner, and Lovász, and the proof relies on this surprising connection to graph theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A characterization for positive semi-definite matrix products Garbe, Frederik Wei, Fan Combinatorics Spectral Theory 05C35, 15A45, 05C50 A well-known fact in linear algebra is that $A^T A$ is always positive semi-definite for any real matrix $A$. We consider a generalization of this fact via the following decision problem. Given a symbolic product of length $k$, consisting of $\ell$ variables and their transposes, such as $ABB^TCA^T$, does there exist an $n\in\mathbb N$ and an assignment of matrices from $\mathbb R^{n\times n}$ such that the resulting matrix product has a negative eigenvalue? We show that this problem is decidable and provide a simple characterization of those symbolic products that have only non-negative real eigenvalues for any assignment of matrices. This characterization can also be understood as a matrix analogue of the positive graph conjecture by Camarena, Csóka, Hubai, Lippner, and Lovász, and the proof relies on this surprising connection to graph theory. |
| title | A characterization for positive semi-definite matrix products |
| topic | Combinatorics Spectral Theory 05C35, 15A45, 05C50 |
| url | https://arxiv.org/abs/2605.02314 |