On Matrix Commutators of Higher Order

Abstract

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the followingProposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.