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arXiv · 2510.20689

The trace Cayley-Hamilton theorem

Darij Grinberg
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Abstract

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.

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Darij Grinberg. 2025-10-23. The trace Cayley-Hamilton theorem. https://arxiv.org/abs/2510.20689

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