The eigenvector variety of a matrix pencil

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×a)-matrices, or, equivalently, by n linear transformations αi:ka→kb with i=1,…,n. We say that P is reduced provided the intersection of the kernels of the linear transformations αi is zero. If P is a reduced matrix pencil, a vector v∈ka will be called an eigenvector of P provided the subspace 〈α1(v),…,αn(v)〉 of kb generated by the elements α1(v),…,αn(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 P(ka), 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.