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The trace Cayley-Hamilton theorem

This expository paper presents new proofs for various properties of matrix traces, determinants, and adjugate matrices over commutative rings, including a demonstration of the trace Cayley-Hamilton theorem.

Original authors: Darij Grinberg

Published 2026-04-15

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Darij Grinberg

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, complex machine made of gears and levers. In mathematics, this machine is a matrix (a grid of numbers). For a long time, mathematicians knew a few "magic rules" about how these machines behave, but they were often complicated, required the machine to be made of specific materials (like real numbers), or relied on looking at the machine's "internal energy levels" (eigenvalues) which don't always exist in simpler worlds.

In this paper, author Darij Grinberg acts like a master mechanic who has found a new, universal tool to understand these machines. He proves a set of rules that work for any machine, no matter what it's made of, without needing to peek inside its engine.

Here is the breakdown of his discoveries, explained with everyday analogies:

1. The Main Characters: The "Trace" and the "Determinant"

Before we get to the big rules, we need to know two key stats about our machine:

The Determinant: Think of this as the machine's "Volume" or "Scale Factor." If you run the machine on a shape, the determinant tells you how much the shape grows or shrinks. If the determinant is zero, the machine has collapsed the shape into nothing (it's broken).
The Trace: Think of this as the machine's "Total Energy" or "Sum of its Core Parts." It's simply the sum of the numbers running down the diagonal of the grid.

2. The Old Rule: The Cayley-Hamilton Theorem

For a long time, mathematicians knew a famous rule called the Cayley-Hamilton Theorem.

The Analogy: Imagine you write a "recipe" (a polynomial) based on the machine's volume (determinant). The rule says: "If you follow this recipe using the machine itself as the ingredients, the machine will stop moving."
In Math: If you plug the matrix $A$ into its own characteristic equation, the result is zero. It's like a machine that, when fed its own blueprint, shuts itself down.

3. The New Rule: The "Trace" Cayley-Hamilton Theorem

This is the paper's main star. The old rule tells us the machine stops. But what if we want to know about the machine's energy (Trace) at every step?

Grinberg proves a deeper relationship:

The Analogy: Imagine the machine is a car. The old rule says, "If you drive according to this map, you will eventually stop." The new rule says, "Not only will you stop, but the sum of your speed (Trace) at every mile marker is perfectly balanced against the map's instructions."
The Formula: The paper shows a specific equation where the Trace of the machine's powers ( $A, A^2, A^3...$ ) is mathematically locked to the coefficients of the machine's "recipe" (the characteristic polynomial).
Why it matters: Usually, to prove this, you need to assume the machine is made of "real" numbers and has "eigenvalues" (like specific frequencies it vibrates at). Grinberg proves this works even if the machine is made of weird, abstract materials (like integers or polynomials) where those frequencies don't exist. He does it without ever looking at the "internal energy levels."

4. The Secret Weapon: The "Magic Deformation" Trick

How did he prove this? He uses a clever trick he calls the "$tI + A$" strategy.

The Problem: Sometimes, a machine is "stuck" or "singular" (its volume is zero). In math, you can't divide by zero, so you can't cancel things out to prove your point.
The Trick: Imagine you have a broken toy car. You can't fix it directly. But, what if you put the car on a moving conveyor belt (adding $tI$)?
- Suddenly, the car is moving! The "volume" of the car-plus-conveyor-belt is no longer zero; it's a perfect, non-zero polynomial.
- Now you can do all your calculations, cancel out terms, and prove your rules easily because the "conveyor belt" ensures nothing gets stuck.
- Once the math is done, you simply turn off the conveyor belt (set $t=0$ ). The rules you proved for the moving version automatically apply to the original, broken machine.
The Metaphor: It's like proving a law of physics works for a falling rock by first proving it works for a rock falling in a wind tunnel, then removing the wind. The core truth remains.

5. What Else Did He Find?

Using this "conveyor belt" trick, Grinberg also proved several other useful facts:

The Adjugate: He showed how to calculate the "inverse" of a machine (the adjugate) using a simple recipe of powers of the machine itself.
Block Machines: He proved rules for how to calculate the volume of a giant machine made by gluing four smaller machines together.
Nilpotency: He answered a question about "broken" machines that eventually stop working completely (become zero). He showed that if the "energy" (Trace) of the machine is zero at every step, the machine is guaranteed to eventually break down completely.

Summary

This paper is a masterclass in algebraic flexibility. Grinberg takes a complex problem that usually requires "smooth" materials (like real numbers) and solves it using a clever "deformation" trick that works for any material.

He essentially says: "Don't try to fix the broken machine directly. Put it on a conveyor belt, solve the problem while it's moving, and then take it off. The solution will still be valid."

This makes the "Trace Cayley-Hamilton Theorem" accessible to everyone, showing that deep mathematical truths hold together even in the most abstract and rigid environments.

Based on the provided paper by Darij Grinberg, here is a detailed technical summary of "The trace Cayley-Hamilton theorem."

1. Problem Statement

The paper addresses two primary objectives within the context of linear algebra over commutative rings (where eigenvalues may not exist):

The Trace Cayley-Hamilton Theorem: To provide a rigorous, elementary proof of the identity relating the coefficients of the characteristic polynomial to the traces of powers of a matrix. Specifically, for an $n \times n$ matrix $A$ with characteristic polynomial $\chi_A(t) = \sum_{i=0}^n c_{n-i}t^i$ , the theorem states:
$k c_k + \sum_{i=1}^k \text{Tr}(A^i) c_{k-i} = 0$
for every $k \in \mathbb{N}$ . While known to experts (often via eigenvalues over fields or universal identities), the paper aims to provide a self-contained proof valid for arbitrary commutative rings without relying on eigenvalues.
Adjugate Properties and Applications: To explore and prove various properties of the adjugate matrix ( $\text{adj } A$ ) and demonstrate a specific algebraic strategy ("the $tI_n + A$ trick") for proving determinant identities that are difficult to establish directly when the determinant is a zero-divisor.

2. Methodology

The paper employs a unified algebraic strategy centered on polynomial matrices and evaluation homomorphisms.

The " $tI_n + A$ " Strategy: Instead of working directly with a matrix $A \in K^{n \times n}$ , the author replaces it with the polynomial matrix $M(t) = tI_n + A \in (K[t])^{n \times n}$ .
- Key Advantage: The determinant $\det(tI_n + A)$ is a monic polynomial of degree $n$ in $K[t]$ . Monic polynomials are regular elements (non-zero-divisors) in the polynomial ring $K[t]$ .
- Mechanism: This regularity allows for the cancellation of $\det(tI_n + A)$ from equations (e.g., $X \cdot \det(M) = Y \cdot \det(M) \implies X=Y$ ), a step that is generally impossible if $\det(A)$ is a zero-divisor in $K$ .
- Conclusion: Once an identity is proven in the ring $K[t]$ using this regularity, the author applies the evaluation homomorphism $\varepsilon: K[t] \to K$ (where $\varepsilon(p) = p(0)$ ) to map the result back to the original ring $K$ .
Adjugate Expansion: The core of the proof for the Cayley-Hamilton theorems involves expanding the adjugate of the characteristic matrix as a polynomial in $t$ :
$\text{adj}(tI_n - A) = \sum_{k=0}^{n-1} t^k D_k$
where $D_k$ are constant matrices in $K^{n \times n}$ .
Derivations: The paper utilizes the concept of $K$ -derivations (specifically the differentiation operator $\partial$ ) to relate the derivative of the characteristic polynomial to the trace of the adjugate matrix.

3. Key Contributions and Results

A. The Trace Cayley-Hamilton Theorem (Theorem 2.6)

The paper provides a new, elementary proof of the trace identity.

Derivation: By differentiating the identity $(tI_n - A)\text{adj}(tI_n - A) = \chi_A(t)I_n$ , the author establishes that $\partial \chi_A = \text{Tr}(\text{adj}(tI_n - A))$ .
Coefficient Matching: Comparing coefficients of powers of $t$ in the derivative of $\chi_A$ and the trace of the expanded adjugate yields a relationship between the coefficients $c_k$ and $\text{Tr}(D_k)$ .
Result: Combining this with the standard Cayley-Hamilton proof (which relates $D_k$ to powers of $A$ ), the author derives the trace identity $k c_k + \sum \text{Tr}(A^i)c_{k-i} = 0$ .

B. Properties of the Adjugate Matrix

The paper proves several classical and non-trivial properties of the adjugate over commutative rings using the $tI_n + A$ strategy:

Product Rule (Theorem 5.10): $\text{adj}(AB) = \text{adj}(B)\text{adj}(A)$ .
Adjugate of Adjugate (Theorem 5.12):
- $\det(\text{adj } A) = (\det A)^{n-1}$ .
- $\text{adj}(\text{adj } A) = (\det A)^{n-2}A$ (for $n \ge 2$ ).
Polynomial Representation (Theorem 5.14): The adjugate is a polynomial in $A$ :
$\text{adj } A = (-1)^{n-1} \sum_{i=0}^{n-1} c_{n-1-i} A^i$
Jacobi's Theorem (Theorem 5.22): A formula relating minors of $\text{adj } A$ to minors of $A$ .
Trace of Adjugate (Theorem 5.32): $\text{Tr}(\text{adj } A) = (-1)^{n-1} c_{n-1} = (-1)^{n-1} [t^1]\chi_A$ .

C. Applications to Block Matrices

The strategy is applied to derive determinant formulas for block matrices where commutativity conditions exist:

Commuting Blocks (Theorem 5.27): If $AC = CA$, then $\det \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det(AD - CB)$ .
Rank-1 Perturbations (Theorem 5.34): A generalized formula for the determinant of a block matrix involving vectors and row/column perturbations, generalizing the Matrix Determinant Lemma.

D. Nilpotency and Traces

Nilpotency Criterion (Corollary 4.1): If $\text{Tr}(A^i) = 0$ for all $1 \le i \le n$ , then $n! A^n = 0$ . If $K$ is a $\mathbb{Q}$ -algebra, then $A^n = 0$ .
Converse (Corollary 4.2): If $\chi_A = t^n$ , then $\text{Tr}(A^i) = 0$ for all $i > 0$ .
Almkvist's Theorem (Section 6.1): The paper outlines a proof (based on exterior algebra concepts but rewritten) showing that if $A^{k+1}=0$ , then $(\text{Tr } A)^{nk+1} = 0$ .

4. Significance

Accessibility: The paper makes the "Trace Cayley-Hamilton theorem" accessible to students and researchers by avoiding advanced machinery like exterior algebras or universal splitting algebras, relying instead on elementary polynomial manipulations.
Rigorous Generalization: It demonstrates that results often proven only for fields (using eigenvalues) hold for arbitrary commutative rings, provided one uses the correct algebraic tools (regularity of monic polynomials).
Unified Technique: The " $tI_n + A$ " trick is presented as a powerful, generalizable method for proving determinant identities. It effectively "deforms" a singular matrix into a regular one, allowing algebraic cancellations that are otherwise forbidden.
Historical Context: The author clarifies the historical attribution of the theorems, noting that while Cayley and Hamilton are associated with the standard theorem, the trace version has roots in Frobenius and Gantmacher, and the specific proof technique (expanding the adjugate) dates back to Buchheim (1884).

In summary, Grinberg's paper serves as both a pedagogical resource for proving the trace Cayley-Hamilton theorem and a technical reference for advanced properties of adjugates and block determinants over commutative rings, unified by a robust algebraic deformation strategy.