On the dimension of affine subspaces of nilpotent matrices

TL;DR

This paper investigates the maximum possible dimensions of affine subspaces of nilpotent matrices with fixed rank, providing specific results for cases where the rank is either one or just below full rank.

Contribution

It offers new bounds and characterizations for the dimensions of affine subspaces of nilpotent matrices at critical rank values, especially for rank 1 and rank n-1.

Findings

Determined maximal dimensions for affine subspaces with rank n-1.

Established bounds for affine subspaces with rank 1.

Provided structural insights into nilpotent matrix subspaces.

Abstract

The focus of the paper is on the maximal dimension of affine subspaces of nilpotent $n \times n $ matrices with fixed rank. In particular we obtain two results in the "border" cases rank equal to $n-1$ and rank equal to $1$.