Frontiers of Mathematics in China

, Volume 14, Issue 5, pp 967–987 | Cite as

Standard tensor and its applications in problem of singular values of tensors

  • Qingzhi YangEmail author
  • Yiyong Li
Research Article


In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.


Standard tensor nonnegative rectangular tensor singular value geometrically simple 


74B99 15A18 15A69 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank two referees for their valuable comments and suggestions, which greatly help improve the paper. This work was supported in part by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2017D01A14).


  1. 1.
    Bomze I M, Ling C, Qi L, Zhang X. Standard bi-quadratic optimization problems and unconstrained polynomial reformulations. J Global Optim, 2012, 52: 663–687MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chang K C, Pearson K, Zhang T. Perron Frobenius Theorem for nonnegative tensors. Commun Math Sci, 2008, 6(2): 507–520MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl, 2011, 32(3): 806–819MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chang K C, Pearson K, Zhang T. Some variational principles for Z-eigenvalues of non-negative tensors. Linear Algebra Appl, 2013, 438(11): 4166–4182MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chang K C, Qi L, Zhang T. A survey on the spectral theory of nonnegative tensors. Numer Linear Algebra Appl, 2013, 20(6): 891–912MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chang K C, Qi L, Zhou G. Singular values of a real rectangular tensor. J Math Anal Appl, 2010, 370: 284–294MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang K C, Zhang T. Multiplicity of singular values for tensors. Commun Math Sci, 2009, 7(3): 611–625MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dahl G, Leinaas J M, Myrheim J, Ovrum E. A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl, 2007, 420: 711–725MathSciNetCrossRefGoogle Scholar
  9. 9.
    De Lathauwer L, De Moor B, Vandewalle J. On the best rank-1 and rank-(R 1,R 2,..., R N) approximation of higher-order tensors. SIAM J Matrix Anal Appl, 2000, 21(4): 1324–1342MathSciNetCrossRefGoogle Scholar
  10. 10.
    Friedland S, Gaubert S, Han L. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438(2): 738–749MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hu S, Huang Z, Qi L. Strictly nonnegative tensors and nonnegative tensor partition. Sci China Math, 2014, 57(1): 181–195MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hu S, Qi L. Algebraic connectivity of an even uniform hypergraph. J Comb Optim, 2012, 24: 564–579MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proc of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. 2005, 129–132Google Scholar
  14. 14.
    Lim L H. Multilinear pagerank: measuring higher order connectivity in linked objects. The Internet: Today and Tomorrow, 2005Google Scholar
  15. 15.
    Ling C, Zhang X, Qi L. Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints. Numer Linear Algebra Appl, 2011, 19: 113–131MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrix Anal Appl, 2009, 31(3): 1090–1099MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ni Q, Qi L, Wang F. An eigenvalue method for the positive definiteness identification problem. IEEE Trans Automat Control, 2008, 53(5): 1096–1107MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pearson K. Essentially positive tensors. Int J Algebra, 2010, 4: 421–427MathSciNetzbMATHGoogle Scholar
  19. 19.
    Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40(6): 1302–1324MathSciNetCrossRefGoogle Scholar
  20. 20.
    Qi L, Dai H -H, Han D. Conditions for strong ellipticity and M-eigenvalues. Front Math China, 2009, 4(2): 349–364MathSciNetCrossRefGoogle Scholar
  21. 21.
    Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China, 2007, 2(4): 501–526MathSciNetCrossRefGoogle Scholar
  22. 22.
    Qi Y, Comon P, Lim L H. Uniqueness of nonnegative tensor approximations. IEEE Trans Inform Theory, 2016, 62(4): 2170–2183MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ragnarsson S, Van Loan C F. Block tensors and symmetric embeddings. Linear Algebra Appl, 2013, 438(2): 853–874MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yang Q, Yang Y. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32(4): 1236–1250MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yang Y, Yang Q. Further results for PerronCFrobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31(5): 2517–2530MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front Math China, 2011, 6(2): 363–378MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yang Y, Yang Q. A note on the geometric simplicity of the spectral radius of non-negative irreducible tensor. arXiv: 1101.2479Google Scholar
  28. 28.
    Zhang X, Ling C, Qi L. Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. J Global Optim, 2010, 49: 293–311MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsKashi UniversityKashiChina
  2. 2.School of Mathematical SciencesNankai UniversityTianjinChina

Personalised recommendations