Positive definiteness and semi-definiteness of even order symmetric cauchy tensors

Haibin Chen, Liqun Qi

Research output: Journal article publicationJournal articleAcademic researchpeer-review

40 Citations (Scopus)


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. 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.
Original languageEnglish
Pages (from-to)1263-1274
Number of pages12
JournalJournal of Industrial and Management Optimization
Issue number4
Publication statusPublished - 1 Jan 2015


  • Cauchy tensor
  • Eigenvalue
  • Generating vector
  • Positive definiteness
  • Positive semi-definiteness

ASJC Scopus subject areas

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

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