The trace Cayley-Hamilton theorem

TL;DR

This paper explores properties of matrix traces, determinants, and the trace Cayley-Hamilton theorem, providing proofs that illustrate linear algebra techniques over commutative rings.

Contribution

It presents proofs of the trace Cayley-Hamilton theorem and related properties, highlighting methods in linear algebra over commutative rings.

Findings

The trace Cayley-Hamilton theorem holds for matrices over commutative rings.

Various properties of matrix traces, determinants, and adjugates are proved.

Some proofs of known results are provided, illustrating general techniques.

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.