9,203 research outputs found
Properties of Some Classes of Structured Tensors
In this paper, we extend some classes of structured matrices to higher order
tensors. We discuss their relationships with positive semi-definite tensors and
some other structured tensors. We show that every principal sub-tensor of such
a structured tensor is still a structured tensor in the same class, with a
lower dimension. The potential links of such structured tensors with
optimization, nonlinear equations, nonlinear complementarity problems,
variational inequalities and the nonnegative tensor theory are also discussed.Comment: arXiv admin note: text overlap with arXiv:1405.1288 by other author
Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors
Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we
further introduce skew centrosymmetric tensors and centrosymmetric Cauchy
tensors, and discuss properties of these three classes of structured tensors.
Some sufficient and necessary conditions for a tensor to be centrosymmetric or
skew centrosymmetric are given. We show that, a general tensor can always be
expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric
tensor. Some sufficient and necessary conditions for a Cauchy tensor to be
centrosymmetric or skew centrosymmetric are also given. Spectral properties on
H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and
centrosymmetric Cauchy tensors are discussed. Some further questions on these
tensors are raised
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
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