  We study the problem of when a periodic square matrix of order $n$ over an
arbitrary field $\mathbb{F}$ 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 $\frac n2$ when $\mathbb{F}$ 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 $\mathbb{F}$ 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.
