# Decomposition of matrices into a sum of invertible matrices and matrices of fixed index

## TL;DR

This paper establishes necessary and sufficient conditions for expressing any nonzero square matrix as a sum of an invertible matrix and a nilpotent matrix of fixed index over any field.

## Contribution

It provides a complete characterization of matrices that can be decomposed into an invertible and a nilpotent matrix with a specified nilpotency index.

## Key findings

- Characterization of matrices as sums of invertible and nilpotent matrices.
- Conditions depend on the size of the matrix and the nilpotency index.
- Results hold over arbitrary fields.

## Abstract

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 $\mathbb{F}$.