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

Abstract: We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to Šter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).

• The paper establishes that every matrix over a finite field of characteristic 2 with more than three elements can be expressed as the sum of a diagonalizable matrix and a square-zero matrix.
• It employs rational canonical decomposition and companion matrices to explicitly construct the diagonalizable and nilpotent parts, demonstrating tight potency bounds.
• The results resolve longstanding questions in matrix decomposition and pave the way for further research in computational linear algebra and finite field theory.

Decomposition of Matrices Over Finite Fields of Characteristic 2

Background and Problem Context

The paper addresses the structural decomposition of square matrices over finite fields with characteristic 2. Specifically, it analyzes the conditions under which any matrix $A \in M_n(F)$, with $F$ a finite field of characteristic 2, can be represented as a sum of a diagonalizable matrix and a nilpotent matrix of index at most 2 (that is, a square-zero matrix). This issue connects to classical investigations into nil-clean rings and the representation of matrices over finite fields as combinations of idempotent and nilpotent elements, prominently initiated in [BCDM], and more recent results on the decomposition into potent and nilpotent matrices.

Prior research established such decompositions for certain field cardinalities, notably $\mathbb{F}_2$, where all matrices can be written as sums of idempotent and nilpotent matrices only if the field has precisely two elements. However, the extension of these results to larger fields, specifically of characteristic 2 and cardinality at least 4, remained unresolved, motivating the present comprehensive treatment.

Main Results and Methodology

The central result is: Every matrix over a finite field of characteristic 2 with cardinality at least 4 can be expressed as the sum of a diagonalizable matrix and a square-zero matrix. This is formally stated in Theorem~\ref{main}, and the proof makes extensive use of rational canonical decomposition, examining companion matrices of invariant factors across all possible degrees.

For $F=\mathbb{F}_2$, the decomposition as sum of a diagonalizable and square-zero is not always possible. The authors show, however, that every matrix can be decomposed as a sum of a potent matrix (with index not exceeding 4) and a nilpotent matrix (index not exceeding 2), thus paralleling results from [St1] and [Sh].

Technical Approach

The authors proceed by:

• Explicitly constructing diagonalizable and square-zero matrices for companion matrices of degree 2, 3, and 4, which form the basis of rational canonical decomposition.
• Demonstrating, via explicit algebraic constructions, that for any non-derogative matrix (minimal polynomial equals characteristic polynomial), such decompositions can be applied recursively to form the overall decomposition of any matrix.
• Extending to polynomials of general degree by leveraging block matrix arguments and direct sum decompositions.
• Adjusting potency indices based on the characteristic polynomials and their coefficients, showing the minimum possible potency order for the decomposition depending on the structure of the matrix.

Strong claims include the assertion that for finite fields of characteristic 2 with more than three elements, every matrix admits the desired decomposition without exception, settling prior open questions. For $\mathbb{F}_2$, the index of potency is at most 4, which cannot, in general, be reduced.

Numerical and Structural Results

The decompositions are fully constructive, with explicit forms for both diagonalizable and square-zero (or nilpotent) components given for each matrix size. Notably:

• For $F = \mathbb{F}_{2^m}$ ($m \geq 2$), the diagonalizable summand is always $2^m$-potent; for $F=\mathbb{F}_2$, all companion matrices of degree $\geq 3$ require a 4-potent diagonalizable component, and degree 2 admits a 2-potent decomposition.
• The decomposition holds for all matrix sizes, with no exceptional cases for $F$0 of cardinality strictly greater than three elements.
• The theoretical lower bounds on potency are proved tight for certain matrices, particularly for non-derogative ones.

Implications and Future Directions

These results complete the picture of the nil-clean and potent decomposition of matrix rings over finite fields of characteristic 2. Practically, the decomposition facilitates matrix analysis, simplifying spectral investigations, and is relevant for ring-theoretic properties such as automorphism groups and module decompositions.

Theoretically, the approach suggests further study into:

• The interplay between field structure (cardinality, characteristic) and possible matrix decompositions.
• The extension to other classes of fields, such as local fields or infinite fields of characteristic 2, where similar canonical decompositions could be attempted.
• Potential generalizations to non-associative or non-linear algebra contexts, such as group algebras or Lie algebras over finite fields.

Finally, in computational settings, the constructive proofs enable algorithmic decomposition—beneficial for symbolic computation and finite field linear algebra.

Conclusion

The paper resolves the longstanding problem of decomposing matrices over finite fields of characteristic 2 into the sum of diagonalizable and square-zero matrices for all fields with more than three elements. For $F$1, decomposition is possible with higher potency (at most 4). The results are comprehensive, structurally explicit, and establish tight potency bounds. The implications are significant for both theoretical linear algebra and computational matrix theory, offering a foundation for further investigations into matrix decompositions over finite fields.