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 $$\mathbb S={\mathbf x\in\mathbb P<sup>n\;|\;\sum\limits_{i=0}<sup>nx_i<sup>2=0}$$. 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 $\mathbb P<sup>n$. 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.