53 research outputs found
Tensor Transpose and Its Properties
Tensor transpose is a higher order generalization of matrix transpose. In
this paper, we use permutations and symmetry group to define? the tensor
transpose. Then we discuss the classification and composition of tensor
transposes. Properties of tensor transpose are studied in relation to tensor
multiplication, tensor eigenvalues, tensor decompositions and tensor rank
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
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
Asymptotics of degrees and ED degrees of Segre products
Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface
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