A combinatorial proof of the trace Cayley-Hamilton theorem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911257463881728 |
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| author | Bera, Sudip |
| author_facet | Bera, Sudip |
| contents | The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed graph D(A), where the algebraic behavior of A is reflected in the combinatorial properties of D(A). In particular, the determinant and characteristic polynomial of A admit elegant formulations in terms of sign-weighted sums over linear subdigraphs of D(A), thereby providing a graphical interpretation of fundamental algebraic quantities. Building upon this correspondence, we establish a combinatorial proof of the trace Cayley-Hamilton theorem. This theorem furnishes explicit trace identities linking the coefficients of the characteristic polynomial of A with the traces of its successive powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06689 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A combinatorial proof of the trace Cayley-Hamilton theorem Bera, Sudip Combinatorics The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed graph D(A), where the algebraic behavior of A is reflected in the combinatorial properties of D(A). In particular, the determinant and characteristic polynomial of A admit elegant formulations in terms of sign-weighted sums over linear subdigraphs of D(A), thereby providing a graphical interpretation of fundamental algebraic quantities. Building upon this correspondence, we establish a combinatorial proof of the trace Cayley-Hamilton theorem. This theorem furnishes explicit trace identities linking the coefficients of the characteristic polynomial of A with the traces of its successive powers. |
| title | A combinatorial proof of the trace Cayley-Hamilton theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.06689 |