矩阵 pencil 的特征向量集
The eigenvector variety of a matrix pencil
AI总结：
本文研究了矩阵 pencil 的特征向量集，证明任何射影概形均可作为某 reduced 矩阵 pencil 的特征向量集。
摘要：
设 $k$ 为一个域，$n,a,b$ 为自然数。给定一个矩阵 pencil $P$，由 $n$ 个相同大小的矩阵组成，系数在 $k$ 中，即由 $(b\times a)$-矩阵，或等价地由 $n$ 个线性变换 $α_i\:k^a \to k^b$ 组成。若 $P$ 是 reduced 矩阵 pencil，则向量 $v\in k^a$ 称为 $P$ 的特征向量，当由 $α_1(v),\dots,α_n(v)$ 生成的子空间 $k^b$ 是 1 维的。特征向量称为等价当它们是标量倍数关系。等价类的集合 $ε(P)$ 是射影空间 $\Bbb P(k^a)$ 的 Zariski 关闭子集，因此是一个射影概形。本文旨在证明任何射影概形均可作为某 reduced 矩阵 pencil 的特征向量集。
英文摘要：
Let $k$ be a field and $n,a,b$ natural numbers. A matrix pencil $P$ is given by $n$ matrices of the same size with coefficients in $k$, say by $(b\times a)$-matrices, or, equivalently, by $n$ linear transformations $α_i\:k^a \to k^b$ with $1=1,\dots,n$. We say that $P$ is reduced provided the intersection of the kernels of the linear transformations $α_i$ is zero. If $P$ is a reduced matrix pencil, a vector $v\in k^a$ will be called an eigenvector of $P$ provided the subspace $\langle α_1(v),\dots,α_n(v) \rangle$ of $k^b$ generated by the elements $α_1(v),\dots,α_n(v)$ is $1$-dimensional. Eigenvectors are called equivalent provided they are scalar multiples of each other. The set $ε(P)$ of equivalence classes of eigenvectors of $P$ is a Zariski closed subset of the projective space $\Bbb P(k^a)$, thus a projective variety. We call it the eigenvector variety of $P$. The aim of this note is to show that any projective variety arises as an eigenvector variety of some reduced matrix pencil.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
