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

Peter Danchev, Esther García, Miguel Gómez Lozano

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 $\mathbb{F}$.

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

Abstract

For any and fixed , we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring to be written as a sum of an invertible matrix and a nilpotent matrix with over an arbitrary field .
Paper Structure (2 sections, 4 theorems, 25 equations)

This paper contains 2 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Decomposing Matrices into a Sum of Invertible and Nilpotent Matrices

Key Result

Lemma 2.2

Let $\mathbb{F}$ be a field, let $n\ge 2$, and let us fix an index of nilpotence $k$ with $2\le k\le n$. For each $1\le r,s\le n$ such that $s+k-2\le n$, the matrices $N_{r,s,k}$ have rank equal to $k-1$ and are nilpotent of index $k$.

Theorems & Definitions (12)

  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • ...and 2 more