Jordan structures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size

TL;DR

This paper characterizes all nilpotent matrices that commute with a specific nilpotent matrix composed of two equal-sized Jordan blocks, detailing their Jordan forms and partitions.

Contribution

It provides a complete classification of nilpotent matrices in the centralizer of a nilpotent matrix with two equal Jordan blocks, including their Jordan canonical forms.

Findings

List of all possible Jordan canonical forms for commuting nilpotent matrices.

Characterization of nilpotent orbits intersecting the centralizer.

Partition-based description of the commuting matrices' structures.

Abstract

In this paper we characterize all nilpotent orbits under the action by conjugation that intersect the nilpotent centralizer of a nilpotent matrix $B$ consisting of two Jordan blocks of the same size. We list all the possible Jordan canonical forms of the nilpotent matrices that commute with $B$ by characterizing the corresponding partitions.