🔢 mathematics

On the principal minors of the powers of a matrix

Inspired by a 2021 Putnam problem, this paper proves that the diagonal entries of any power of a matrix are uniquely determined by and can be expressed as universal integral polynomials of the matrix's principal minors, implying that if all principal minors equal one, so do the diagonal entries of all powers.

Original authors: Darij Grinberg

Published 2026-07-01

📖 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

The Big Picture: The "Fingerprint" of a Matrix

Imagine you have a complex machine, which we'll call Matrix A. This machine is a grid of numbers (an $n \times n$ matrix).

In the world of mathematics, there is a special way to take a "snapshot" of this machine. Instead of looking at every single number inside, you look at specific, self-contained chunks of it. You take a chunk by picking a set of rows and the exact same set of columns. The "snapshot" of that chunk is its determinant (a single number calculated from that smaller grid).

These snapshots are called Principal Minors.

The smallest snapshots are just single numbers on the diagonal (1x1 chunks).
The largest snapshot is the whole machine itself (the determinant of the entire matrix).
There are $2^n$ possible snapshots in total.

The paper asks a very specific question: If you know all the snapshots of Machine A, do you know everything about Machine A?

The answer is no. The paper shows that two different machines can have the exact same set of snapshots but behave differently when you run them twice (square them). It's like two different people having the same fingerprints but different voices; the fingerprints don't tell you everything.

The Breakthrough: The "Diagonal" Clue

However, the paper discovers a very specific part of the machine that is fully determined by those snapshots.

Imagine the machine has a row of lights running diagonally from the top-left to the bottom-right. These are the diagonal entries.

The paper proves a surprising fact: If you know all the principal minors (the snapshots) of Machine A, you can perfectly predict the numbers on the diagonal of Machine A squared, cubed, or raised to any power.

It's as if the "shadows" cast by the machine (the minors) contain enough information to reconstruct the "core lights" (the diagonal) of the machine after it has been run through a process multiple times.

How It Works: The "Universal Recipe"

The author doesn't just say "it's possible"; he shows how it's possible.

He proves that there is a Universal Recipe (a mathematical formula made of whole numbers) for every single diagonal light.

If you want to know the top-left light of the machine after it's been run 5 times ( $A^5$ ), you don't need to know the original machine's internal wiring. You just need to plug the values of the original snapshots into this specific recipe.
This recipe works for any machine, whether it's made of integers, fractions, or even abstract algebraic objects. The recipe is "universal."

The Analogy:
Think of the principal minors as the ingredients in a soup (salt, pepper, carrots, etc.).

Usually, knowing the ingredients doesn't tell you exactly what the soup tastes like after you cook it for 10 hours (the power $m$ ).
But, this paper proves that for the specific flavor of the "center of the bowl" (the diagonal), there is a magic formula. If you know the exact amount of every ingredient (the minors), you can calculate the center flavor of the soup after any amount of cooking time, using a formula that never changes.

The Special Case: The "All-Ones" Machine

The paper also explores a special scenario. Imagine a machine where every single snapshot (principal minor) equals 1.

The paper proves that if you take this machine and run it through the process any number of times (square it, cube it, etc.), the lights on the diagonal will always remain 1.

The Analogy:
Imagine a magical kaleidoscope where every pattern you see is perfectly symmetrical (value = 1). The paper proves that no matter how many times you twist the kaleidoscope (raise the matrix to a power), the center of the view will always stay perfectly symmetrical.

Why This Matters (In the Context of the Paper)

This research was inspired by a famous math competition problem (Putnam 2021) that asked about "odd" numbers. The official solution to that problem was tricky and relied on specific number properties.

Grinberg's paper steps back and says: "Let's look at the algebraic structure." He shows that the relationship between the snapshots and the diagonal is so strong that it holds true universally, not just for odd numbers.

He also addresses a lingering mystery: Does this rule hold for every part of the machine, or just the diagonal?

The Verdict: Just the diagonal. The off-diagonal parts (the other numbers) are a bit more chaotic; they depend on slightly more complex "quasi-shadows" that aren't just simple snapshots.

Summary

The Problem: You can't always predict a matrix's future behavior just by looking at its principal minors (snapshots).
The Discovery: You can perfectly predict the diagonal numbers of the matrix's powers ( $A^m$ ) using a universal formula based solely on those snapshots.
The Special Case: If all snapshots are 1, the diagonal numbers of the powered matrix will always be 1.
The Method: The proof uses a clever mathematical trick involving "formal power series" (think of them as infinite polynomials) to connect the snapshots to the diagonal lights, showing that the connection is unbreakable.

The paper is a triumph of pure algebra, showing that even when a system seems too complex to predict, there are specific, hidden patterns (the diagonal) that are rigidly locked to the system's fundamental properties (the minors).

Technical Summary: On the Principal Minors of the Powers of a Matrix

Problem Statement
The paper addresses a question arising from Problem B5 of the Putnam contest 2021, which established that if all principal minors of an $n \times n$ matrix $A$ over $\mathbb{Z}$ are odd, then all principal minors of any power $A^m$ are also odd. While the original problem is solved, the algebraic mechanism behind this phenomenon remains opaque. Specifically, it is unclear whether principal minors of a matrix power $A^m$ are uniquely determined by the principal minors of the base matrix $A$ . The author notes that this is generally false: Example 1.2 demonstrates that two matrices $C$ and $D$ can share identical principal minors yet possess different principal minors in their squares ( $C^2$ and $D^2$ ), depending on specific off-diagonal interactions.

Despite this non-uniqueness for general principal minors, the paper investigates whether the diagonal entries of $A^m$ (which are themselves $1 \times 1$ principal minors) are uniquely determined by the principal minors of $A$ .

Methodology
The author employs a universal algebraic approach combined with formal power series techniques.

Universal Polynomial Construction: To prove that diagonal entries of $A^m$ are determined by the principal minors of $A$ , the author constructs a "general" $n \times n$ matrix $A$ over a polynomial ring $R = \mathbb{Z}[x_{i,j}]$ . By proving that the diagonal entries of $A^m$ lie in the subring generated by the principal minors of this generic matrix, the result extends to any commutative ring via algebra homomorphisms.
Formal Power Series and Adjugates: The core proof utilizes the matrix $B = I_n - tA$ within the ring of formal power series $R[[t]]$ . The inverse of this matrix is given by the geometric series $B^{-1} = I_n + tA + t^2A^2 + \dots$ . The coefficient of $t^m$ in the $(i,i)$ -entry of $B^{-1}$ corresponds exactly to the $(i,i)$ -entry of $A^m$ .
Determinantal Identities: The proof leverages the identity $B^{-1} = (\det B)^{-1} \cdot \text{adj}(B)$ $B^{- 1} = (det B)^{- 1} \cdot adj (B)$ . The author expands $\det B$ $det B$ and the diagonal entries of the adjugate matrix $\text{adj}(B)$ $adj (B)$ using the formula for the determinant of a sum involving a scalar multiple of the identity matrix (a known folklore result relating characteristic polynomials to principal minors).
- It is shown that $\det B$ is a power series with coefficients in the subring $P$ generated by the principal minors of $A$ , and its constant term is 1, making it invertible in $P[[t]]$ .
- Similarly, the diagonal entries of $\text{adj}(B)$ are shown to be power series with coefficients in $P$ .
- Consequently, the product $(\det B)^{-1} \cdot (\text{adj}(B))_{i,i}$ , which equals $(B^{-1})_{i,i}$ , has coefficients in $P$ .

Key Contributions and Results

Theorem 2.1 (Universal Polynomials): For any nonnegative integers $n, m$ and index $i$ , there exists an integer polynomial $P_{n,i,m}$ in $2^n$ variables such that the $i$ -th diagonal entry of $A^m$ is obtained by substituting the principal minors of $A$ into this polynomial. This confirms that diagonal entries of powers are uniquely determined by the principal minors of the base matrix.
Theorem 2.2 (Subring Membership): Let $P$ be the subring of a commutative ring $R$ generated by all principal minors of $A$ . Then, all diagonal entries of $A^m$ belong to $P$ . This is the algebraic engine behind Theorem 2.1.
Theorem 5.1 (Off-Diagonal Variant): The paper extends the analysis to off-diagonal entries $(A^m)_{i,j}$ . It introduces the concept of " $(i,j)$ -quasiprincipal minors" (determinants of submatrices where row and column sets differ only by swapping $i$ and $j$ ). It proves that $(A^m)_{i,j}$ belongs to the module generated by products of principal minors and these quasiprincipal minors.
Theorem 5.2 (Unit Minors): If all principal minors of $A$ equal 1, then all diagonal entries of $A^m$ equal 1. The proof shows that under this assumption, $\det(I_n - tA) = (1-t)^n$ and $(\text{adj}(I_n - tA))_{i,i} = (1-t)^{n-1}$ , leading to $(B^{-1})_{i,i} = (1-t)^{-1} = \sum t^k$ , where the coefficient of $t^m$ is 1.

Significance and Scope
The paper clarifies the algebraic structure underlying the behavior of matrix powers regarding principal minors. It distinguishes between the general case (where principal minors of $A^m$ are not determined by those of $A$ ) and the specific case of diagonal entries (which are determined).

The author explicitly states that these results shed "new light on the behavior of minors under matrix multiplication" and provide a polynomial framework for the Putnam 2021 result. However, the paper remains modest regarding broader generalizations:

It does not claim that all principal minors of $A^m$ are determined by those of $A$ (citing Example 1.2 as a counterexample).
It leaves open the question (Question 6.1) of whether the condition "principal minors equal 1" implies "all principal minors of $A^m$ equal 1" for arbitrary rings. The author notes that while the answer is positive for integral domains, a counterexample exists for certain quotient rings of polynomial rings, and the case for $R = \mathbb{Z}/4$ remains open (though an update notes a proof for $\mathbb{Z}/d$ was found by an LLM).

The work is presented as a technical resolution to the specific algebraic mechanism of the Putnam problem, offering universal polynomials and subring membership proofs without extending into experimental applications or unverified future implications.