On the nilpotent commutator of a nilpotent matrix

Abstract

We study the structure of the nilpotent commutator 𝒩 _B of a nilpotent matrix B. We show that 𝒩 _B intersects all nilpotent orbits for conjugation if and only if B is a square-zero matrix. We describe nonempty intersections of 𝒩 _B with nilpotent orbits in the case the n × n matrix B has rank n − 2. Moreover, we give some results concerning the inverse image of the map taking B to the maximal nilpotent orbit intersecting 𝒩 _B .

Keywords:

• nilpotent matrix
• commuting matrices
• nilpotent commutator
• nilpotent orbit
• maximal partition

AMS Subject Classifications::

• 15A27
• 14L30
• 15A21