On the nilpotent commutator of a nilpotent matrix

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.

arXiv comment14 pages