505 research outputs found
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition
Hankel tensors arise from applications such as signal processing. In this
paper, we make an initial study on Hankel tensors. For each Hankel tensor, we
associate it with a Hankel matrix and a higher order two-dimensional symmetric
tensor, which we call the associated plane tensor. If the associated Hankel
matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel
tensor. We show that an order -dimensional tensor is a Hankel tensor if
and only if it has a Vandermonde decomposition. We call a Hankel tensor a
complete Hankel tensor if it has a Vandermonde decomposition with positive
coefficients. We prove that if a Hankel tensor is copositive or an even order
Hankel tensor is positive semi-definite, then the associated plane tensor is
copositive or positive semi-definite, respectively. We show that even order
strong and complete Hankel tensors are positive semi-definite, the Hadamard
product of two strong Hankel tensors is a strong Hankel tensor, and the
Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We
show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd
order) are nonnegative. We give some upper bounds and lower bounds for the
smallest and the largest Z-eigenvalues of a Hankel tensor, respectively.
Further questions on Hankel tensors are raised
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
Convergence of a Second Order Markov Chain
In this paper, we consider convergence properties of a second order Markov
chain. Similar to a column stochastic matrix is associated to a Markov chain, a
so called {\em transition probability tensor} of order 3 and dimension
is associated to a second order Markov chain with states. For this ,
define as on the dimensional standard simplex
. If 1 is not an eigenvalue of on and is
irreducible, then there exists a unique fixed point of on . In
particular, if every entry of is greater than , then 1 is not
an eigenvalue of on . Under the latter condition, we
further show that the second order power method for finding the unique fixed
point of on is globally linearly convergent and the
corresponding second order Markov process is globally -linearly convergent.Comment: 16 pages, 3 figure
Infinite and finite dimensional Hilbert tensors
For an -order dimensional Hilbert tensor (hypermatrix)
, its
spectral radius is not larger than , and an upper
bound of its -spectral radius is . Moreover,
its spectral radius is strictly increasing and its -spectral radius is
nondecreasing with respect to the dimension . When the order is even, both
infinite and finite dimensional Hilbert tensors are positive definite. We also
show that the -order infinite dimensional Hilbert tensor (hypermatrix)
defines a bounded and
positively -homogeneous operator from into (),
and the norm of corresponding positively homogeneous operator is smaller than
or equal to
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