Hankel tensors: Associated Hankel matrices and Vandermonde decomposition

Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

50 Citations (Scopus)


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 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 m order n-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-eigenvalues 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.
Original languageEnglish
Pages (from-to)113-125
Number of pages13
JournalCommunications in Mathematical Sciences
Issue number1
Publication statusPublished - 1 Jan 2014


  • Co-positiveness
  • Eigenvalues of tensors
  • Generating functions
  • Hankel matrices
  • Hankel tensors
  • Plane tensors
  • Positive semi-definiteness
  • Vandermonde decomposition

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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