Trace Cayley-Hamilton Theorem

• The trace Cayley-Hamilton theorem is a fundamental linear algebra result that connects the coefficients of a matrix’s characteristic polynomial with the traces of its powers.
• It employs adjugate expansions and algebraic differentiation to extend classical Newton–Girard identities to arbitrary commutative rings without relying on eigenvalue analysis.
• Its implications include criteria for nilpotency, advancements in polynomial identity theory, and applications in settings like modular arithmetic and invariant theory.

The trace Cayley–Hamilton theorem is a foundational result in linear algebra that refines the classical Cayley–Hamilton theorem by connecting the coefficients of the characteristic polynomial of a square matrix with the traces of its powers. This theorem extends across arbitrary commutative rings and provides a bridge between determinantal invariants and power sums, generalizing classical Newton identities in a robust, eigenvalue-free fashion. Its significance is magnified in contexts where spectral theory does not apply, and its techniques highlight the deep interplay between the trace, determinant, and adjugate matrix.

1. The Trace Cayley–Hamilton Theorem: Statement and Scope

Let $A$ be an $n \times n$ matrix over a commutative ring $\mathbb{K}$, and let its characteristic polynomial be expressed as

$$\chi_A(t) = \det(tI_n - A) = \sum_{k=0}^n c_{n-k} t^k,$$

where $c_0 = 1$ under the convention of monicity. The trace Cayley–Hamilton theorem asserts that for every $k \in \mathbb{N}$,

$$k \, c_k + \sum_{i=1}^k \operatorname{Tr}(A^i) c_{k-i} = 0,$$

where $\operatorname{Tr}(A^i)$ denotes the trace of the $i$-th power of $A$ (Grinberg, 23 Oct 2025).

This identity holds in full generality over any commutative $\mathbb{K}$, independent of the existence of eigenvalues, and generalizes the familiar trace-determinant relationships valid over fields.

2. Algebraic Framework and Proof Techniques

The approach to establishing the trace Cayley–Hamilton theorem utilizes only algebraic and combinatorial constructs without reliance on eigenvectors or field extensions.

• The key step is to consider $A$ as a constant matrix in the polynomial ring $\mathbb{K}[t]$ with associated $n \times n$ structure $tI_n - A$.
• The adjugate matrix, defined entrywise as

$$(\operatorname{adj} X)_{i,j} = (-1)^{i+j} \det(X_{\sim j, \sim i}),$$

where $X_{\sim j, \sim i}$ is the submatrix of $X$ with row $j$ and column $i$ deleted, is central to the argument.

• The adjugate polynomial matrix admits an expansion

$$\operatorname{adj}(tI_n - A) = D_0 + D_1 t + \cdots + D_{n-1} t^{n-1},$$

with $D_k$ matrices in $\mathbb{K}^{n \times n}$ depending polynomially (in a highly structured way) on $A$.

A crucial identity is then utilized: $$\frac{d}{dt} \chi_A(t) = \operatorname{Tr}(\operatorname{adj}(tI_n - A)),$$ which equates the derivative of the determinant to the trace of the adjugate. Coefficient extraction and comparison, combined with the basic properties of determinants and adjugates (such as $X \operatorname{adj} X = (\det X) I_n$), yield the desired formula without recourse to spectral theory.

3. Interplay with Newton–Girard and Classical Invariants

The formula

$$k \, c_k + \sum_{i=1}^k \operatorname{Tr}(A^i) c_{k-i} = 0$$

is, up to sign, the noncommutative analogue of the classical Newton–Girard identities, which relate elementary symmetric functions of eigenvalues (the $c_k$) to the power sums (traces of powers).

Whereas for diagonalizable matrices these connect sums over eigenvalues and products of eigenvalues, in the non-diagonalizable or non-field context, the traces of powers and the characteristic polynomial coefficients remain polynomially determined by the ring operations. The significance of this generalization is that it enables "trace-to-determinant" transfer even when direct spectral decomposition is not possible.

In particular, the trace Cayley–Hamilton theorem allows recovery of the entire characteristic polynomial from the traces of the first $n$ powers of $A$ (and vice versa), providing an explicit system of linear equations among the fundamental polynomial invariants of $A$.

4. Role of Adjugate, Trace, and Determinant

Three central matrix functions are woven together in the theorem:

• Determinant: The function $\det(tI_n - A)$ is the polynomial generator whose coefficients are the focus of the theorem. Over non-fields, the notion is entirely algebraic and is constructed via the universal property of the determinant.
• Trace: The trace $\operatorname{Tr}(A^i)$ serves as the power sum invariant, computable as the sum of diagonal elements rather than as a sum of eigenvalue powers, and maintains its integrity in contexts where eigenvalues may not exist.
• Adjugate: The adjugate's expansion in powers of $t$ and its properties (e.g., adj$(tI_n - A) \cdot (tI_n - A) = \chi_A(t) I_n$) furnish the foundational algebraic manipulations. The explicit expression of the derivative of the determinant in terms of the trace of the adjugate is a key step in the direct proof.

These elements act together to avoid the necessity of eigenvalue decompositions and instead connect matrix invariants via pure ring-theoretic identities.

5. Applications and Consequences

Several immediate implications and applications arise from the trace Cayley–Hamilton theorem:

• Nilpotency Criteria: If $\operatorname{Tr}(A^i) = 0$ for $1 \leq i \leq n$, then the theorem implies that all $c_k$ vanish (up to characteristic), so the characteristic polynomial is $t^n$; consequently, $A^n = 0$ or, in positive characteristic, $(n!)A^n = 0$, depending on invertibility of $n!$.
• Polynomial Identity Rings: The theorem provides essential tools in the paper of rings and algebras satisfying polynomial identities, feeding into both PI theory and invariant theory since trace identities generate central elements and/or help control the structure of polynomial invariants.
• Determinantal Extensions: The approach of expanding the adjugate and differentiating the determinant revisits and extends numerous classical determinant identities, all holding in the arbitrary commutative ring context.
• Commutative Ring Linear Algebra: The methodology is optimal in settings where eigenvalues may not be defined, or diagonalizability fails. For example, in modular arithmetic or rings of dual numbers, all trace-determinant relations remain valid.

6. Divergence from Traditional (Spectral) Methods

Classical derivations of trace-elementary symmetric relations employ spectral theory, relying on eigenvector decompositions and manipulation of eigenvalues—tools unavailable over general rings. The trace Cayley–Hamilton theorem, as proved in (Grinberg, 23 Oct 2025), avoids all spectral arguments, replacing them by manipulation of adjugate expansions and algebraic differentiation.

Additionally, the proof techniques rely on the universal applicability of the ring homomorphism $\varepsilon:\mathbb{K}[t]\to \mathbb{K}$ defined by evaluation at $t=0$ and the invertibility of monic polynomials (e.g., $\det(tI_n+A)$ is always a monic power of $t$), allowing algebraic cancellations that circumvent issues with non-invertible determinants.

7. Broader Impact and Advanced Interconnections

The trace Cayley–Hamilton theorem underpins numerous further developments:

• In the context of generalized or functional matrix identities (e.g., quasi-identities, central polynomials), this result forms the backbone for deducing trace-based identities as consequences of the characteristic polynomial, as explored in (Brešar et al., 2012).
• In noncommutative PI-algebra or Hopf algebra settings, the role of the trace Cayley–Hamilton theorem is evident in the construction of discriminant ideals and representation-theoretic invariants (Huang et al., 27 Jun 2025), as well as in the paper of Poisson trace orders (Brown et al., 2022).
• In geometry and combinatorics, variants and extensions appear, for instance, in hypermatrix generalizations and their applications to graph invariants (Gnang, 2014).

The theorem’s methods offer an instructive model: leveraging algebraic properties of adjugate and trace, one constructs universal identities among matrix invariants, circumventing analytic or spectral dependencies. This generality both preserves and extends the reach of classical linear algebra into abstract and non-field settings.