41 research outputs found
Tensor Complementarity Problem and Semi-positive Tensors
The tensor complementarity problem (\q, \mathcal{A}) is to
\mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q +
\mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) =
0. We prove that a real tensor is a (strictly) semi-positive
tensor if and only if the tensor complementarity problem (\q, \mathcal{A})
has a unique solution for \q>\0 (\q\geq\0), and a symmetric real tensor is
a (strictly) semi-positive tensor if and only if it is (strictly) copositive.
That is, for a strictly copositive symmetric tensor , the tensor
complementarity problem (\q, \mathcal{A}) has a solution for all \q \in
\mathbb{R}^n
Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors
and their generating vectors in this paper. Hilbert tensors are symmetric
Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite
if and only if its generating vector is positive. An even order symmetric
Cauchy tensor is positive definite if and only if its generating vector has
positive and mutually distinct entries. This extends Fiedler's result for
symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that
the positive semi-definiteness character of an even order symmetric Cauchy
tensor can be equivalently checked by the monotone increasing property of a
homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial
is strictly monotone increasing in the nonnegative orthant of the Euclidean
space when the even order symmetric Cauchy tensor is positive definite.
Furthermore, we prove that the Hadamard product of two positive semi-definite
(positive definite respectively) symmetric Cauchy tensors is a positive
semi-definite (positive definite respectively) tensor, which can be generalized
to the Hadamard product of finitely many positive semi-definite (positive
definite respectively) symmetric Cauchy tensors. At last, bounds of the largest
H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and
several spectral properties on Z-eigenvalues of odd order symmetric Cauchy
tensors are shown. Further questions on Cauchy tensors are raised
Three Dimensional Strongly Symmetric Circulant Tensors
In this paper, we give a necessary and sufficient condition for an even order
three dimensional strongly symmetric circulant tensor to be positive
semi-definite. In some cases, we show that this condition is also sufficient
for this tensor to be sum-of-squares. Numerical tests indicate that this is
also true in the other cases