Linear operators and positive semidefiniteness of symmetric tensor spaces

Zi Yan Luo, Liqun Qi, Yin Yu Ye

Research output: Journal article publicationJournal articleAcademic researchpeer-review

17 Citations (Scopus)


We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor. Furthermore, we characterize the symmetric positive semidefinite tensor (SDT) cone by employing the properties of linear operators, design some face structures of its dual cone, and analyze its relationship to many other tensor cones. In particular, we show that the cone is self-dual if and only if the polynomial is quadratic, give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases, and develop a complete relationship map among the tensor cones appeared in the literature.
Original languageEnglish
Pages (from-to)197-212
Number of pages16
JournalScience China Mathematics
Issue number1
Publication statusPublished - 1 Jan 2015


  • linear operator
  • SOS cone
  • symmetric positive semidefinite tensor cone
  • symmetric tensor

ASJC Scopus subject areas

  • Mathematics(all)


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