Maximal dimension of affine subspaces of specific matrices

Abstract

For every $n \in \mathbb{N}$ and every field $K$, let $N(n,K)$ be the set of the nilpotent $n \times n$ matrices over $K$ and let $D(n,K) $ be the set of the $n \times n$ matrices over $K$ which are diagonalizable over $K$. Moreover, let $R(n) $ be the set of the normal $n \times n$ matrices. In this short note we prove that the maximal dimension of an affine subspace in $N(n,K)$ is $ \frac{n(n-1)}{2}$ and, if the characteristic of the field is zero, an affine not linear subspace in $N(n,K)$ has dimension less than or equal to $ \frac{n(n-1)}{2}-1$. Moreover we prove that the maximal dimension of an affine subspace in $R(n)$ is $n$, the maximal dimension of a linear subspace in $D(n, \mathbb{R})$ is $ \frac{n(n+1)}{2}$, while the maximal dimension of an affine not linear subspace in $D(n, \mathbb{R})$ is $ \frac{n(n+1)}{2} -1$.