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

Abstract

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.