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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.20689 |
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| _version_ | 1866910129343954944 |
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| author | Grinberg, Darij |
| author_facet | Grinberg, Darij |
| contents | In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the *trace Cayley-Hamilton theorem*, which says that \[ kc_k + \sum_{i=1}^k \operatorname{Tr} (A^i) c_{k-i} = 0 \qquad \text{for every } k\in\mathbb{N} \] whenever $A$ is an $n\times n$-matrix with characteristic polynomial $\det (tI_n - A) = \sum_{i=0}^n c_{n-i} t^i$ over a commutative ring $\mathbb{K}$. While the results are not new, some of the proofs are. The proofs illustrate some general techniques in linear algebra over commutative rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20689 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The trace Cayley-Hamilton theorem Grinberg, Darij Rings and Algebras History and Overview 15A15, 15A24 In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the *trace Cayley-Hamilton theorem*, which says that \[ kc_k + \sum_{i=1}^k \operatorname{Tr} (A^i) c_{k-i} = 0 \qquad \text{for every } k\in\mathbb{N} \] whenever $A$ is an $n\times n$-matrix with characteristic polynomial $\det (tI_n - A) = \sum_{i=0}^n c_{n-i} t^i$ over a commutative ring $\mathbb{K}$. While the results are not new, some of the proofs are. The proofs illustrate some general techniques in linear algebra over commutative rings. |
| title | The trace Cayley-Hamilton theorem |
| topic | Rings and Algebras History and Overview 15A15, 15A24 |
| url | https://arxiv.org/abs/2510.20689 |