论文标题
具有恒定等级的反对称矩阵的仿射子空间
Affine subspaces of antisymmetric matrices with constant rank
论文作者
论文摘要
对于\ mathbb {n} $和每个字段$ k $中的每一个$ n \，让$ a（n，k）$是反对称$（n \ times n）$ - 矩阵上$ k $的矢量空间。我们说，如果每个$ s $的每个矩阵都有排名$ r $，则仿射子空间$ s $ $ a（n，k）$具有恒定等级$ r $。定义$$ {\ cal a} _ {antisym}^k（n; r）= \ {s \; | | \; s \; \ mbox {\ rm affine子空间的$ a（n，k）的常数等级} r \} $$ $$ $ $ $ $ a_ {antisym}^k（n; r）= \ max \ {\ max \ {\ dim s \ s \ s \ in mid s \ mid s \ in {\ cal a}美元$$ a_ {antisym}^{\ mathbb {r}}（n; 2r）= r（r+1）。$$
For every $n \in \mathbb{N}$ and every field $K$, let $A(n,K)$ be the vector space of the antisymmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $A(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}_{antisym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subspace of $A(n,K)$ of constant rank } r\}$$ $$a_{antisym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{antisym}^K(n;r) \}.$$ In this paper we prove the following formulas: for $n \geq 2r +2 $ $$a_{antisym}^{\mathbb{R}}( n; 2r) = (n-r-1) r ;$$ for $n=2r$ $$a_{antisym}^{\mathbb{R}}( n; 2r) =r(r-1) ;$$ for $n=2r+1$ $$a_{antisym}^{\mathbb{R}}( n; 2r) = r(r+1) .$$
