The E-Eigenvectors of Tensors

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 $\mathbb{S}=\{\mathbf{x}\in {\mathbb{P}}^n|\sum _{i=0}^n{x}_i^2=0\}$. 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.

17 pages

References in corpus (1)

• Introduction to Non-Linear Algebra

Cited by in corpus (1)

• The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph