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Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices

This paper resolves the problem of decomposing square matrices over finite fields of characteristic 2 into the sum of a diagonalizable matrix and a square-zero matrix for all fields with more than three elements, while also establishing a related decomposition for matrices over $\mathbb{F}_2$ involving a potent matrix.

Original authors: Peter Danchev, Esther García, Miguel Gómez Lozano

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Peter Danchev, Esther García, Miguel Gómez Lozano

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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. This machine represents a matrix (a grid of numbers) used in math and computer science. The paper you're asking about is a recipe for taking apart any such machine, no matter how complicated, and rebuilding it using only two very simple, specific types of parts.

Here is the breakdown of the paper's discovery, translated into everyday language with some creative analogies.

The Big Goal: The "Two-Part" Puzzle

The mathematicians (Peter, Esther, and Miguel) wanted to answer a specific question: Can every possible matrix be built by adding together just two special kinds of matrices?

They wanted to split any matrix $A$ into:

The "Steady" Part (Diagonalizable): Think of this as a machine that runs smoothly and predictably. If you look at it from the right angle, it's just a straight line of gears turning at constant speeds. It never gets stuck or chaotic.
The "Flash" Part (Square-Zero): This is a machine that works for one split second and then immediately stops forever. If you run it twice ( $N \times N$ ), it completely vanishes into nothingness. It's a "one-hit wonder" that leaves no trace after the second use.

The Question: Can we take any complex matrix and say, "Hey, this is just a Steady Part plus a Flash Part"?

The Setting: The World of "Even" Numbers

This research takes place in a very specific universe called a Finite Field of Characteristic 2.

The Analogy: Imagine a world where the only numbers are 0 and 1, and the rule is that $1 + 1 = 0$ . It's like a light switch that is either Off (0) or On (1). If you turn it on twice, it turns off. This is the math behind how computers process data (binary).
The paper looks at fields that are slightly bigger than just 0 and 1 (like having 4, 8, 16, etc., distinct "colors" of switches), but they all follow that "addition is like flipping a switch" rule.

The Main Discovery: "Yes, but with a Twist"

The paper solves this puzzle for almost all these worlds.

1. The Easy Worlds (Fields with 4 or more elements)

If the field has at least 4 different "colors" (numbers), the answer is a resounding YES.

The Result: Every single matrix in these worlds can be perfectly split into a Steady Part and a Flash Part.
The Metaphor: It's like saying, "No matter how messy your room is, you can always organize it by putting all the clothes in the closet (Steady) and throwing all the trash in the bin (Flash) so that the trash disappears after you step on it once."
Why it matters: This settles a long-standing debate in mathematics. Before this, people weren't sure if this was possible for all sizes of matrices in these fields. The authors proved it works for every size.

2. The Tricky World (The Field with only 2 elements: $\mathbb{F}_2$ )

This is the world with only 0 and 1. It's the smallest, most restrictive world.

The Problem: In this tiny world, you can't always find a "Steady Part" that is perfectly diagonal. Sometimes the "Steady" part gets stuck.
The Solution: The authors found a clever workaround. They proved that in this tiny world, you can still break the matrix down, but the "Steady Part" needs to be slightly more flexible. Instead of being perfectly steady, it just needs to be a "Potent" Part.
The Analogy: Imagine the "Steady Part" is a dancer. In the big worlds, the dancer does a perfect, endless loop. In the tiny world, the dancer might do a loop, stop, and then repeat the exact same loop four times before stopping. It's not perfectly smooth, but it's predictable enough.
The Result: Every matrix in the 0-and-1 world can be split into a Flash Part (vanishes after 2 uses) and a Potent Part (repeats its pattern every 4 uses).

Why Should You Care? (The "So What?")

You might wonder, "Who cares about splitting grids of numbers?"

Simplifying Complexity: In computer science and engineering, we often deal with massive, messy data matrices. Knowing that any complex system can be broken down into a "predictable" part and a "temporary" part helps engineers design better algorithms. It's like knowing that any complicated song is just a melody (steady) plus a drum beat that fades out (flash).
Solving Old Mysteries: For years, mathematicians had examples of matrices that couldn't be split this way in the tiny 0-and-1 world. This paper didn't just say "it's impossible"; it said, "It's impossible exactly as you asked, but here is a slightly different, equally powerful way to do it." They fixed the puzzle by changing the rules just a tiny bit.
The "Magic" of Characteristic 2: This research highlights how the weird rules of binary math (where $1+1=0$ ) create unique challenges and unique solutions that don't exist in normal math.

Summary in One Sentence

The authors proved that in the binary world of computers, any complex mathematical structure can be dismantled into a predictable, repeating pattern and a temporary burst of activity, provided we allow the pattern to repeat a few times in the smallest possible version of that world.

They didn't just solve a math problem; they showed us that even in the most restrictive, binary universe, chaos can always be tamed by breaking it down into simple, understandable pieces.

1. Problem Statement

The paper addresses the decomposition problem for square matrices over finite fields of characteristic 2. Specifically, it investigates whether any matrix $A \in M_n(\mathbb{F})$ can be expressed as the sum:
$A = D + N$
where:

$D$ is a diagonalizable matrix (or a potent matrix, i.e., $D^t = D$ for some $t \geq 2$ ).
$N$ is a square-zero matrix (nilpotent of index at most 2, i.e., $N^2 = 0$ ).

Context and Motivation:

Previous work established that over the field $\mathbb{F}_2$ , not every matrix is the sum of an idempotent ( $D^2=D$ ) and a square-zero matrix. Counterexamples were provided by Šter and Shitov.
Breaz (2018) posed the question of whether matrices over finite fields of even cardinality $q \geq 4$ admit a decomposition into a $q$ -potent matrix and a square-zero matrix.
The authors aim to resolve this completely for all finite fields of characteristic 2, distinguishing between fields with $|\mathbb{F}| \geq 4$ and the specific case of $\mathbb{F}_2$ .

2. Methodology

The authors employ a structural approach based on the Rational Canonical Form (or Frobenius Normal Form). Since any square matrix is similar to a direct sum of companion matrices of its invariant factors, the problem reduces to proving the decomposition for non-derogative matrices (matrices where the minimal polynomial equals the characteristic polynomial).

The methodology proceeds in two main phases:

Phase A: Fields with Cardinality $\geq 4$ ( $|\mathbb{F}| \geq 4$ )

The authors construct explicit decompositions for non-derogative matrices of various orders ( $n$ ) by analyzing the trace and the degree of the characteristic polynomial.

Small Orders ( $n=2, 3, 4$ ): They define specific families of diagonalizable matrices ( $D_2, D_3, D_4$ ) and square-zero matrices ( $N_3, N_4$ ) with carefully chosen eigenvalues. They prove that any non-derogative matrix of these orders can be written as $N + D$ where $D$ has distinct eigenvalues in $\mathbb{F}$ .
General Orders ( $n \geq 5$ ): They use an inductive block-matrix construction.
- For orders $n = 4k$ , $4k+1$ , $4k+2$ , and $4k+3$ , they construct a square-zero matrix $N$ such that $A - N$ becomes a block-diagonal matrix $D$ .
- The matrix $D$ is constructed as a direct sum of smaller diagonalizable blocks (orders 2, 3, and 4) defined in the earlier propositions.
- The construction relies on the existence of an element $a \in \mathbb{F}$ such that $a \neq 0, 1$ . Since $|\mathbb{F}| \geq 4$ , such an element always exists.
- They verify that the constructed $N$ satisfies $N^2=0$ and that $D$ is diagonalizable by explicitly providing a basis of eigenvectors.

Phase B: The Field $\mathbb{F}_2$

Since $\mathbb{F}_2$ lacks an element $a \notin \{0, 1\}$ , the previous construction fails. The authors adapt the method by working over the extension field $\mathbb{F}_4$ (where $a^2+a+1=0$ ) and then showing that the resulting decomposition matrices $N$ and $D$ actually have entries in $\mathbb{F}_2$ .

They prove that for $\mathbb{F}_2$ , the diagonalizable part $D$ is not necessarily idempotent ( $D^2=D$ ), but is 4-potent ( $D^4=D$ ).
This is achieved by analyzing the specific forms of companion matrices over $\mathbb{F}_2$ and utilizing the properties of $\mathbb{F}_4$ to ensure the resulting $D$ has eigenvalues in $\{0, 1, a, a+1\}$ , which satisfy $x^4=x$ .

3. Key Contributions and Results

Main Theorem (Theorem 3.1)

For any finite field $\mathbb{F}$ of characteristic 2 with $|\mathbb{F}| \geq 4$ :

Every square matrix $A \in M_n(\mathbb{F})$ can be expressed as $A = D + N$ , where $D$ is diagonalizable and $N$ is square-zero ( $N^2=0$ ).

Corollary 3.2: Since every diagonalizable matrix $D$ over a finite field of size $q=2^m$ satisfies $D^q = D$ , it follows that every matrix over such a field is the sum of a $q$ -potent matrix and a square-zero matrix. This positively answers Breaz's question for all even fields with $q \geq 4$ .

Results for $\mathbb{F}_2$ (Corollary 3.3)

For the field $\mathbb{F}_2$ :

Every square matrix $A \in M_n(\mathbb{F}_2)$ can be expressed as $A = D + N$ , where $N^2=0$ and $D$ is 4-potent ( $D^4=D$ ).

This result is significant because it shows that while the decomposition into an idempotent and a square-zero matrix fails for $\mathbb{F}_2$ (as shown by Šter and Shitov), the decomposition holds if the potency index is relaxed to 4.

Refined Potency Index (Proposition 3.4)

The authors provide conditions under which the index of potency for $D$ can be reduced to the cardinality of a specific subfield $K \subseteq \mathbb{F}$ (where $K \neq \mathbb{F}_2$ ).

If the trace or specific coefficients of the characteristic polynomial lie in a subfield $K$ , the diagonalizable part $D$ can be chosen such that $D^{|K|} = D$ .
For $n=2$ with trace 0, they show $D$ can be chosen such that $D^3=D$ .

4. Significance

Complete Resolution: The paper completely settles the decomposition problem for matrices over finite fields of characteristic 2. It confirms that the obstruction to decomposing into an idempotent and a square-zero matrix is specific to $\mathbb{F}_2$ and the requirement of idempotency.
Generalization of Breaz's Question: It confirms that for all finite fields of even characteristic with $q \geq 4$ , the decomposition $A = D + N$ holds with $D^q = D$ and $N^2 = 0$ .
Refinement for $\mathbb{F}_2$ : It provides a constructive proof that matrices over $\mathbb{F}_2$ are sums of a square-zero matrix and a 4-potent matrix, bridging the gap left by previous counterexamples regarding idempotency.
Constructive Approach: The proofs are highly constructive, providing explicit algorithms (via block matrices and eigenvector bases) to generate the decomposition, rather than relying solely on existence arguments.

Conclusion

The authors successfully demonstrate that the "bad behavior" of matrices over $\mathbb{F}_2$ regarding idempotent decompositions is an artifact of the small field size. By expanding the class of the "diagonalizable" component to include $q$ -potent matrices (or 4-potent matrices for $\mathbb{F}_2$ ), a universal decomposition theorem is established for all finite fields of characteristic 2.