The eigenvector variety of a matrix pencil

Abstract

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.