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Gómez-Lozano, Miguel Ángel García, Esther Danchev, Peter 2025-12-11T11:56:30Z 2025-12-11T11:56:30Z 2025 Peter Danchev, Esther García, Miguel Gómez Lozano, Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices, Linear Algebra and its Applications, Volume 730, 2026, Pages 35-50, ISSN 0024-3795, https://doi.org/10.1016/j.laa.2025.10.002. https://hdl.handle.net/10630/41062 10.1016/j.laa.2025.10.002 Let Fbe a finite field of odd characteristic. When |F|≥5, we prove that every matrix A admits a decomposition into D+M, where Dis diagonalizable and M2=0. For F= F3, we show that such a decomposition is possible for non derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over F3, never admit such a decomposition. These results completely settle the question posed by Breaz (2018) [3] asking if it is true that, for big enough positive integers n≥3, all matrices Aover a field of odd cardinality qadmit decompositions of the form E+Mwith Eq=E and M2=0: specifically, the answer is yes for q≥5, but however there are counterexamples for q =3and each order n =3k, whenever k ≥ 1. eng http://creativecommons.org/licenses/by/4.0/ open access Attribution 4.0 Internacional Cuerpos modulares Determinantes Matrices over finite fields of odd characteristic as sums of diagonalizable and square- zero matrices. journal article