46,936 research outputs found
E-Characteristic Polynomials of Tensors
In this paper, we show that the coefficients of the E-characteristic
polynomial of a tensor are orthonormal invariants of that tensor. When the
dimension is 2, some simplified formulas of the E-characteristic polynomial are
presented. A re- sultant formula for the constant term of the E-characteristic
polynomial is given. We then study the set of tensors with infinitely many
eigenpairs and the set of irregular tensors, and prove both the sets have
codimension 2 as subvarieties in the projective space of tensors. This makes
our perturbation method workable. By using the perturbation method and
exploring the difference between E-eigenvalues and eigenpair equivalence
classes, we present a simple formula for the coefficient of the leading term of
the E-characteristic polynomial, when the dimension is 2
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 . We show that a generic
tensor has no eigenvectors on . Actually, we show that a generic
tensor has no eigenvectors on a proper nonsingular projective variety in
. 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 is nonsingular and symmetric, its
E-eigenvectors are exactly the singular points of a class of hypersurfaces
defined by 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 as irreducible
factors.Comment: 17 page
Limits of space-times in five dimensions and their relation to the Segre Types
A limiting diagram for the Segre classification in 5-dimensional space-times
is obtained, extending a recent work on limits of the energy-momentum tensor in
general relativity. Some of Geroch's results on limits of space-times in
general relativity are also extended to the context of five-dimensional
Kaluza-Klein space-times.Comment: Late
The necessary and sufficient conditions of copositive tensors
In this paper, it is proved that (strict) copositivity of a symmetric tensor
is equivalent to the fact that every principal sub-tensor of
has no a (non-positive) negative -eigenvalue. The
necessary and sufficient conditions are also given in terms of the
-eigenvalue of the principal sub-tensor of the given tensor. This
presents a method of testing (strict) copositivity of a symmetric tensor by
means of the lower dimensional tensors. Also the equivalent definition of
strictly copositive tensors is given on entire space .Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608
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