  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 $\alpha_i\:k^a
\to k^b$ with $1=1,\dots,n$. We say that $P$ is reduced provided the
intersection of the kernels of the linear transformations $\alpha_i$ 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 $\langle \alpha_1(v),\dots,\alpha_n(v)
\rangle$ of $k^b$ generated by the elements $\alpha_1(v),\dots,\alpha_n(v)$ is
$1$-dimensional. Eigenvectors are called equivalent provided they are scalar
multiples of each other. The set $\epsilon(P)$ 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.
