Characteristic polynomial and higher order traces of third order three dimensional tensors

  • Guimei Zhang
  • Shenglong HuEmail author
Research Article


Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.


Tensor traces characteristic polynomial 


15A18 15A69 


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This work was partially supported by the National Natural Science Foundation of China (Grant No. 11171328), the Natural Science Foundation of Zhejiang Province, China (Grant No. LD19A010002), and the Innovation Research Foundation of Tianjin University (Grant Nos. 2017XZC-0084, 2017XRG-0015). The second author was also supported by the Young Elite Scientists Sponsorship Program by Tianjin.


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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsTianjin UniversityTianjinChina
  2. 2.Department of Mathematics, School of ScienceHangzhou Dianzi UniversityHangzhouChina

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