Decompositions of Periodic Matrices into a Sum of Special Matrices

We study the problem of when a periodic square matrix of order 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 n2 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. Moreover, we prove that each periodic square matrix over any field is a sum of an idempotent matrix and a torsion matrix.