Decompositions of periodic matrices into a sum of special matrices

Abstract

We study the problem of when a periodic square matrix of order n×n over an arbitrary field F is decomposable into the sum of a square-zero matrix and a torsion matrix and show that this decomposition can always be obtained for matrices of rank at least n/2 when F is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when F equals the field of the real numbers.