For every $n \in \mathbb{N}$ and every field $K$, let $M(n \times n, K) $ be the set of $n \times n$ matrices over $K$, let $N(n,K)$ be the set of nilpotent $n \times n$ matrices over $K$ and let $D(n,K) $ be the set of $n \times n$ matrices over $K$ which are diagonalizable over $K$, that is, which are diagonalizable in $M(n \times n, K) $. Moreover, if $K$ is a field with an involutory automorphism, let $R(n,K) $ be the set of normal $n \times n$ matrices over $K$. 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 $R(n, \C)$ is $n$, the maximal dimension of a linear subspace in $D(n, \R)$ is $ \frac{n(n+1)}{2}$, while the maximal dimension of an affine not linear subspace in $D(n, \R)$ is $ \frac{n(n+1)}{2} -1$.
Maximal dimension of affine subspaces of specific matrices / Elena Rubei. - In: OPERATORS AND MATRICES. - ISSN 1846-3886. - STAMPA. - .:(In corso di stampa), pp. 1-9.
Maximal dimension of affine subspaces of specific matrices
Elena Rubei
In corso di stampa
Abstract
For every $n \in \mathbb{N}$ and every field $K$, let $M(n \times n, K) $ be the set of $n \times n$ matrices over $K$, let $N(n,K)$ be the set of nilpotent $n \times n$ matrices over $K$ and let $D(n,K) $ be the set of $n \times n$ matrices over $K$ which are diagonalizable over $K$, that is, which are diagonalizable in $M(n \times n, K) $. Moreover, if $K$ is a field with an involutory automorphism, let $R(n,K) $ be the set of normal $n \times n$ matrices over $K$. 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 $R(n, \C)$ is $n$, the maximal dimension of a linear subspace in $D(n, \R)$ is $ \frac{n(n+1)}{2}$, while the maximal dimension of an affine not linear subspace in $D(n, \R)$ is $ \frac{n(n+1)}{2} -1$.File | Dimensione | Formato | |
---|---|---|---|
subspacesnilpotentnormalv7.pdf
Accesso chiuso
Tipologia:
Versione finale referata (Postprint, Accepted manuscript)
Licenza:
Tutti i diritti riservati
Dimensione
226.01 kB
Formato
Adobe PDF
|
226.01 kB | Adobe PDF | Richiedi una copia |
I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.