Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

Let F be a finite field of odd characteristic. When | F | ≥ 5 , we prove that every matrix A admits a decomposition into D + M , where D is diagonalizable and M 2 = 0 . For F = F 3 , 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 F 3 , never admit such a decomposition. These results completely settle the question posed by Breaz in Lin. Algebra & Appl. (2018) asking if it is true that, for big enough positive integers n ≥ 3 , all matrices A over a field of odd cardinality q admit decompositions of the form E + M with E q = E and M 2 = 0 : specifically, the answer is yes for q ≥ 5 , but however there are counterexamples for q = 3 and each order n = 3 k , whenever k ≥ 1 .