The trace Cayley-Hamilton theorem

TL;DR

This paper provides an eigenvalue-free, algebraic development of the Cayley–Hamilton theorem and the trace-Cayley–Hamilton theorem for matrices over arbitrary commutative rings. It centers on expanding the adjugate $\operatorname{adj}(tI_n-A)$ as a polynomial in $t$, linking its coefficients to the characteristic polynomial coefficients $c_j$, and deriving a derivative identity $\partial \chi_A = \operatorname{Tr}(\operatorname{adj}(tI_n-A))$. From these ingredients, it proves CH and the trace form $k c_k + \sum_{i=1}^k \operatorname{Tr}(A^i) c_{k-i}=0$, then derives nilpotency criteria and a suite of adjugate-related results, including functoriality, the evaluation map, products, adjugates of adjugates, and Jacobi’s theorem on minors. The approach provides a transparent, structurally rich pathway to determinant and trace identities without eigenvalue methods, with broad applicability to linear algebra over rings and to representation-theoretic and combinatorial contexts.

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.

Theorems & Definitions (183)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Corollary 2.4
• Theorem 2.5: Cayley-Hamilton theorem
• Theorem 2.6: trace Cayley-Hamilton theorem
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem.ta+b.prod']}.
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem.poly.1']}.