DECOMPOSITIONS OF MATRICES INTO A SUM OF INVERTIBLE MATRICES AND MATRICES OF FIXED NILPOTENCE

Abstract

For any $n\ge 2$ and fi xed $k\ge 1$, we give necessary and suficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k = 0$ over an arbitrary fi eld $\mathbb{F}$.