On the nilpotent commutator of a nilpotent matrix

TL;DR

This paper investigates the structure of the nilpotent commutator of a nilpotent matrix, characterizing when it intersects all nilpotent orbits and describing intersections for matrices of rank n-2.

Contribution

It provides a complete characterization of when the nilpotent commutator intersects all nilpotent orbits and describes intersections for matrices with rank n-2.

Findings

The nilpotent commutator intersects all nilpotent orbits if and only if the matrix is square-zero.

Descriptions of nonempty intersections for matrices of rank n-2.

Results on the maximal nilpotent orbit intersected by the commutator.

Abstract

We study the structure of the nilpotent commutator $\nb$ of a nilpotent matrix $B$. We show that $\nb$ intersects all nilpotent orbits for conjugation if and only if $B$ is a square--zero matrix. We describe nonempty intersections of $\nb$ with nilpotent orbits in the case the $n \times n$ matrix $B$ has rank $n-2$. Moreover, we give some results on the maximal nilpotent orbit that $\nb$ intersects nontrivially.