Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence

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Peter Danchev
Esther García
Miguel Gómez Lozano


For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient 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 field $\mathbb{F}$.

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