The E-eigenvectors of tensors

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 (Formula presented.). We show that a generic tensor has no eigenvectors on (Formula presented.). Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in (Formula presented.). 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 (Formula presented.) is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by (Formula presented.) 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 (Formula presented.) as irreducible factors.