$n$th Roots of $n$th Powers

This paper explores the optimization of unimodular zerofree matrices as a means to find simple and efficient solutions to a matrix equation involving $n$th roots of $n$th powers.

The Gist

Imagine you are a chef trying to bake a cake. You know the final result: a delicious, perfectly baked cake (let's call it $C^n$). Your challenge is to figure out the exact recipe (the matrix $X$) that, when you follow it $n$ times, produces that cake. In math terms, you are looking for the $n$-th root of a matrix.

This paper, written by Steven Finch, is a culinary adventure through the world of numbers, specifically looking at how to find these "recipes" for 2x2, 3x3, and larger grids of numbers (matrices).

Here is the story broken down into simple, everyday concepts:

1. The Odd vs. The Even (The Weather Forecast)

The author starts by looking at a specific puzzle from 1879 involving a 2x2 grid of numbers. He asks: "If I cube a matrix (multiply it by itself 3 times), how many different recipes could I have used?"

• The Odd Numbers (3, 5, 7...): When the exponent is an odd number, the universe is well-behaved. There are a finite number of solutions. It's like having a specific list of 9 possible recipes. Some are "real" (using normal numbers), and some are "complex" (using imaginary numbers, which are like flavors that don't exist in the real world but are necessary for the math to work).
• The Even Numbers (2, 4, 6...): When the exponent is even, things get chaotic. There are infinitely many solutions. It's as if you could tweak your recipe with a tiny pinch of salt or a drop of water, and it would still work. The author explains this happens because the "ingredients" (eigenvalues) in the even case are identical, causing a collapse in structure, whereas in the odd case, they are distinct and hold the structure together.

2. The Quest for the "Perfect" Recipe

Once the author realized that finding these recipes was possible, he asked a new question: "Which recipe is the simplest?"

In math, "simple" often means keeping the numbers in the recipe as small as possible. If your recipe calls for numbers like 100, 200, and 500, it's messy. If it uses 1, 2, and 3, it's elegant.

The author introduces a special type of recipe book called a Unimodular Zerofree Matrix.

• Unimodular: The recipe is perfectly balanced (the determinant is 1 or -1), meaning it doesn't distort the universe too much.
• Zerofree: This is the tricky part. The recipe cannot contain the number zero. Imagine trying to bake a cake where you are forbidden from using "no eggs" or "no flour." Every single ingredient slot must have something in it.

The goal is to find the matrix $M$ (the transformation tool) that, when used to build a new matrix, results in the smallest possible numbers.

3. The "Goldilocks" Matrices

The author acts like a detective, searching through millions of combinations to find the "Goldilocks" matrix—the one that is just right.

• For 2x2 grids: He found a "perfect" matrix that looks like this:
  $$\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$$
  This is the simplest possible tool that keeps all numbers non-zero and small.
• For 3x3 grids: It gets harder. The "perfect" tool becomes a bit more complex, with numbers like 1, 2, and 3.
• The Surprise (4x4 grids): You might think that as the grid gets bigger, the numbers get bigger and messier. But here's a twist! For 4x4 grids, the author found a solution that is actually more efficient (smaller numbers) than the 3x3 solution. It's like finding a bigger house that is actually cheaper to build than a smaller one.

4. The "Look-Alike" Problem (Canonical Representatives)

One of the biggest headaches in this math is that many recipes look different but are actually the same.

• Imagine you have a recipe card. If you swap the order of the ingredients (swap rows) or flip the sign of an ingredient (multiply by -1), it's still the same cake.
• The author uses a computer program to "canonicalize" these matrices. Think of this as a standardized filing system. No matter how you shuffle the cards or flip the signs, the computer sorts them into a single, standard "ID card."
• This helps him prove that two matrices that look different are actually just the same object wearing a different costume.

5. The Future (The 5x5 and Beyond)

The author then tries to scale this up to 5x5, 6x6, and even 7x7 grids.

• The Challenge: As the grids get bigger, the search becomes like finding a needle in a haystack the size of a galaxy.
• The Discovery: He found examples for 5x5 and 6x6 grids, but they are messy. The numbers get large (like 8, 9, or even 11).
• The Mystery: He suspects there might be a "perfect" 5x5 grid with small numbers (max value of 3), but his computer hasn't found it yet. He also wonders if a 9x9 grid exists that follows these strict "no zero" rules.

The Big Picture

This paper is essentially a search for mathematical elegance. The author wants to find the most efficient, smallest, and cleanest ways to build complex mathematical structures.

He uses computers (like Mathematica and Magma) to brute-force through millions of possibilities, looking for patterns. He discovers that sometimes, bigger dimensions (like 4x4) can be surprisingly simpler than smaller ones (3x3), and he leaves the reader with a challenge: Can you find the perfect 5x5 or 9x9 matrix?

In a nutshell: It's a treasure hunt for the simplest, most elegant "no-zero" number grids that can generate complex mathematical powers, revealing that sometimes the biggest puzzles have the simplest solutions, and sometimes the biggest grids are the hardest to solve.

Technical Summary

Here is a detailed technical summary of the paper "nth Roots of nth Powers" by Steven Finch.

1. Problem Statement

The paper investigates the problem of finding matrix solutions to the equation $X^n = C^n$, where $C$ is a given integer matrix and $n$ is an integer exponent. The primary focus is on identifying simple, efficient, and "zerofree" solutions.

• Context: The study is motivated by historical examples (e.g., 1879 solutions for $X^3 = C^3$) and the observation that while solutions exist, they often involve complex algebraic expressions or large integer entries.
• Goal: To derive "efficient" examples of matrices $A$ (where $A = X^n$) such that the entries of $A$ are minimized. Specifically, the author seeks to minimize the norm $\|A\|$, defined as the maximum absolute value of any entry in the matrix.
• Key Constraint: The author focuses on zerofree matrices, where neither the matrix $M$ nor its inverse $M^{-1}$ contains any zero entries. This constraint is crucial for ensuring the structural integrity and "efficiency" of the resulting solutions.

2. Methodology

The author employs an experimental and computational approach rather than a purely theoretical derivation, acknowledging that the literature may contain undetected examples. The methodology involves:

• Diagonalization Strategy:
  The core technique relies on the spectral decomposition of a target matrix $A$. If $A$ has distinct eigenvalues $\Lambda = \text{diag}(\lambda_1, \dots, \lambda_k)$, then $A$ can be expressed as:
  $$A = M \Lambda M^{-1}$$
  Consequently, the $n$-th roots of $A^n$ can be constructed as:
  $$X_{n, j_1, \dots, j_k} = M \cdot \text{diag}(\lambda_1 \omega^{j_1}, \dots, \lambda_k \omega^{j_k}) \cdot M^{-1}$$
  where $\omega = \exp(2\pi i/n)$ and $j$ are indices ranging from $0$to$n-1$.

• Optimization of the Transformation Matrix ($M$):
  To minimize the entries of the resulting matrix $A$, the author searches for unimodular (determinant $\pm 1$) and zerofree integer matrices $M$. The optimization problems defined are:

  1. Find a unimodular zerofree $M$ minimizing $\|(M, M^{-1})\|$ (the max entry of the concatenated matrix).
  2. Find $M$ such that $\|M \Lambda M^{-1}\|$ is minimal for a given set of eigenvalues.
  3. Find $M$ such that the resulting $A = M \Lambda M^{-1}$ is also zerofree.

• Canonicalization and Equivalence:
  Since many matrices are equivalent under signed permutations (shuffling rows/columns and flipping signs), the author uses computational tools (Magma and Mathematica) to define a "canonical representative" for each orbit. This allows for the elimination of redundant solutions and the rigorous proof of inequivalence between distinct matrix classes.

3. Key Contributions and Results

A. Analysis of $n \times n$ Matrices

The paper systematically explores matrices of increasing dimensions:

• $2 \times 2$ Matrices:

  • The author identifies that for odd $n$, solutions are finite, while for even $n$, solutions can be infinite due to coincident eigenvalues in specific cases (e.g., $C^2 = -3I$).
  • By optimizing the unimodular matrix $M = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ with eigenvalues $\{2, 3\}$, the author derives a "simplest" expression for $X^n = A^n$ with a maximum entry of 4, improving upon previous examples with maximums of 6.
  • The paper proves that two seemingly similar matrices $A$ and $\tilde{A}$ (with the same eigenvalues) are inequivalent under signed permutations, a fact verified via canonical forms.

• $3 \times 3$ Matrices:

  • The search yields 576 solutions for the optimization problem.
  • The optimal unimodular zerofree matrix $M$ is found to be:
    $$M = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 3 \end{pmatrix}$$
  • Using eigenvalues $\{1, 2, 3\}$, this generates a matrix $A$ with a maximum entry of 15. The paper provides explicit formulas for the $n$-th roots of this matrix.

• $4 \times 4$ Matrices:

  • A surprising result is found: The minimal norm for the concatenated matrix $\|(M, M^{-1})\|$ drops to 2 for $4 \times 4$matrices, whereas it was$\ge 3$for$3 \times 3$. This suggests that increasing dimension can sometimes improve "efficiency."
  • Two inequivalent optimal matrices ($M$ and $\tilde{M}$) are identified, each with 129,024 total solutions in their respective orbits.

• Higher Dimensions ($5 \times 5$to$8 \times 8$):

  • The complexity increases significantly. The author provides explicit examples of unimodular zerofree matrices and their inverses for dimensions 5, 6, 7, and 8.
  • For $n=5$, the minimal norm is conjectured to be 3, though finding the specific matrix is computationally difficult.
  • The existence of such matrices is confirmed up to $n=8$. The existence for $n=9$ remains an open question, though the author believes the answer is yes.

B. Canonical Representatives

The paper introduces a rigorous method for classifying matrices under the action of signed permutation groups.

• Using Magma and Mathematica, the author defines a CanonicalizeMatrix function that returns the lexicographically smallest matrix in an orbit.
• This method successfully distinguishes between matrices that appear similar but belong to different equivalence classes (e.g., the $2 \times 2$and$4 \times 4$ cases mentioned above).

C. Addendum on Divergence

In the addendum, the author presents $5 \times 5$and$6 \times 6$examples where the norm of the matrix$M$is strictly less than the norm of its inverse$M^{-1}$(e.g.,$|M|=2$vs$|M^{-1}|=3$). This "divergence" behavior was not observed in the minimal$4 \times 4$ cases, highlighting a new complexity in higher dimensions.

4. Significance

• Pedagogical Value: The paper emphasizes "pedagogical simplicity," aiming to provide clean, integer-based examples of matrix roots that avoid the "lengthy calculations" and complex algebraic expressions often found in standard linear algebra literature.
• Computational Insight: It demonstrates the power of brute-force algorithms combined with group-theoretic canonicalization to solve optimization problems in matrix theory where analytical solutions are elusive.
• Open Problems: The paper establishes a clear frontier for future research, specifically regarding the minimal norms for dimensions $n \ge 5$ and the existence of zerofree unimodular matrices for $n=9$.
• Historical Context: It connects modern computational findings with 19th-century mathematical discoveries, showing how simple integer matrices can generate complex root structures.

In summary, Steven Finch provides a comprehensive computational exploration of finding minimal-norm, zerofree integer matrices that serve as bases for generating $n$-th roots of $n$-th powers, offering both explicit examples and a framework for classifying their equivalence.