Decomposition of a square matrix into a product of a nilpotent matrix and an involutive or idempotent matrix

In this note, we give a complete characterization of square matrices of size $n$ over a commutative field which can be expressed as a product of a nilpotent matrix and an involutive matrix and matrices which can be expressed as a product of a nilpotent matrix and an idempotent matrix, in terms of invariant factors.