29 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
Hypergraphs and hypermatrices with symmetric spectrum
It is well known that a graph is bipartite if and only if the spectrum of its
adjacency matrix is symmetric. In the present paper, this assertion is
dissected into three separate matrix results of wider scope, which are extended
also to hypermatrices. To this end the concept of bipartiteness is generalized
by a new monotone property of cubical hypermatrices, called odd-colorable
matrices. It is shown that a nonnegative symmetric -matrix has a
symmetric spectrum if and only if is even and is odd-colorable. This
result also solves a problem of Pearson and Zhang about hypergraphs with
symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.
Separately, similar results are obtained for the -spectram of
hypermatrices.Comment: 17 pages. Corrected proof on p. 1
The E-Eigenvectors of Tensors
We first show that the eigenvector of a tensor is well-defined. The
differences between the eigenvectors of a tensor and its E-eigenvectors are the
eigenvectors on the nonsingular projective variety . We show that a generic
tensor has no eigenvectors on . Actually, we show that a generic
tensor has no eigenvectors on a proper nonsingular projective variety in
. By these facts, we show that the coefficients of the
E-characteristic polynomial are algebraically dependent. Actually, a certain
power of the determinant of the tensor can be expressed through the
coefficients besides the constant term. Hence, a nonsingular tensor always has
an E-eigenvector. When a tensor is nonsingular and symmetric, its
E-eigenvectors are exactly the singular points of a class of hypersurfaces
defined by and a parameter. We give explicit factorization of the
discriminant of this class of hypersurfaces, which completes Cartwright and
Strumfels' formula. We show that the factorization contains the determinant and
the E-characteristic polynomial of the tensor as irreducible
factors.Comment: 17 page