The trace Cayley-Hamilton theorem

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 $\det (t{I}_n-A)={\sum }_{i=0}^n{c}_{n-i}{t}^i$ 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.

73 pages. Expository paper on linear algebra featuring determinantal identities and folklore techniques. Uploaded here for easier referencing. v2 fixes minor typos and adds Corollary 6.7