4,515 research outputs found
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
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
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 Dominant Eigenvalue of an Essentially Nonnegative Tensor
It is well known that the dominant eigenvalue of a real essentially
nonnegative matrix is a convex function of its diagonal entries. This convexity
is of practical importance in population biology, graph theory, demography,
analytic hierarchy process and so on. In this paper, the concept of essentially
nonnegativity is extended from matrices to higher order tensors, and the
convexity and log convexity of dominant eigenvalues for such a class of tensors
are established. Particularly, for any nonnegative tensor, the spectral radius
turns out to be the dominant eigenvalue and hence possesses these convexities.
Finally, an algorithm is given to calculate the dominant eigenvalue, and
numerical results are reported to show the effectiveness of the proposed
algorithm
On spectral hypergraph theory of the adjacency tensor
We study both and -eigenvalues of the adjacency tensor of a uniform
multi-hypergraph and give conditions for which the largest positive or
-eigenvalue corresponds to a strictly positive eigenvector. We also
investigate when the -spectrum of the adjacency tensor is symmetric
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