The strong nilpotency index of a matrix

TL;DR

This paper investigates the structure of strongly nilpotent matrices over division rings by introducing the strong nilpotency index, and applies these findings to quasi-translation maps with specific nilpotency properties.

Contribution

It characterizes strongly nilpotent matrices using the strong nilpotency index and explores applications to quasi-translation maps with nilpotent Jacobians.

Findings

Strongly nilpotent matrices are linearly triangularizable.

The structure of such matrices can be described via the strong nilpotency index.

Applications to quasi-translation maps with Jacobian of strong nilpotency index two.

Abstract

It is known that strongly nilpotent matrices over a division ring are linearly triangularizable. We describe the structure of such matrices in terms of the strong nilpotency index. We apply our results on quasi-translation x + H such that JH has strong nilpotency index two.