The eigenvector variety of a matrix pencil

TL;DR

This paper introduces the eigenvector variety of a matrix pencil, showing that every projective variety can be realized as such a variety, linking algebraic geometry with matrix pencil theory.

Contribution

It establishes that any projective variety can be represented as the eigenvector variety of a reduced matrix pencil, bridging a gap between algebraic geometry and linear algebra.

Findings

Eigenvector varieties form Zariski closed subsets of projective space.

Any projective variety can be realized as an eigenvector variety.

The concept generalizes classical eigenvector notions to matrix pencils.

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 $\alpha_i\:k^a \to k^b$ 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.