The E-Eigenvectors of Tensors

TL;DR

This paper establishes the well-definedness of tensor eigenvectors, explores their properties on projective varieties, and provides explicit factorizations related to E-eigenvectors and the tensor's discriminant.

Contribution

It introduces the concept of E-eigenvectors, analyzes their existence on projective varieties, and derives explicit factorizations of the discriminant related to tensor eigenvalues.

Findings

Generic tensors have no eigenvectors on certain projective varieties.

A nonsingular tensor always has an E-eigenvector.

The discriminant of hypersurfaces related to symmetric tensors factors into the determinant and E-characteristic polynomial.

Abstract

We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety $\mathbb S=\{\mathbf x\in\mathbb P^n\;|\;\sum\limits_{i=0}^nx_i^2=0\}$. We show that a generic tensor has no eigenvectors on $\mathbb S$. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in $\mathbb P^n$. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor $\mathcal T$ is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by $\mathcal T$ and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces, which completes Cartwright and Strumfels' formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor $\mathcal T$ as irreducible factors.