Skip to main content
SpringerLink
Log in

Strictly nonnegative tensors and nonnegative tensor partition

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We introduce a new class of nonnegative tensors-strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some necessary and sufficient conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. Some preliminary numerical results show the feasibility and effectiveness of the algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berman A, Plemmom R. Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press, 1979

    MATH  Google Scholar 

  2. Chang K C. A nonlinear Krein Rutman theorem. J Syst Sci Complex, 2009, 22: 542–554

    Article  MathSciNet  Google Scholar 

  3. Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507–520

    Article  MATH  MathSciNet  Google Scholar 

  4. Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NZQ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl, 2011, 32: 806–819

    Article  MATH  MathSciNet  Google Scholar 

  5. Chang K C, Pearson K, Zhang T. Some variational principles of the Z-eigenvalues for nonnegative tensors. Linear Algebra Appl, 2013, 438: 4166–4182

    Article  MathSciNet  Google Scholar 

  6. Chang K C, Qi L, Zhang T. A survey on the spectral theory of nonnegative tensors. Numer Linear Algebra Appl, doi: 10.1002/nla.1902, 2013

    Google Scholar 

  7. Chang K C, Zhang T. On the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors. J Math Anal Appl, 2013, 408: 525–540

    Article  MathSciNet  Google Scholar 

  8. Friedland S, Gaubert S, Han L. Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra Appl, 2013, 438: 738–749

    Article  MATH  MathSciNet  Google Scholar 

  9. Gaubert S, Gunawardena J. The Perron-Frobenius theorem for homogeneous, monotone functions. Trans Amer Math Soc, 2004, 356: 4931–4950

    Article  MATH  MathSciNet  Google Scholar 

  10. Hu S L, Qi L. Convergence of a second order Markov chain. ArXiv:1307.6919, 2013

    Google Scholar 

  11. Hu S L, Qi L, Zhang G. The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors. ArXiv:1203.3675, 2012

    Google Scholar 

  12. Lim L H. Singular values and eigenvalues of tensors: A variational approach. In: Computational Advances in Multi-Sensor Adaptive Processing, vol. 1. Puerto Vallarta: IEEE, 2005, 129–132

    Google Scholar 

  13. Liu Y, Zhou G L, Ibrahim N F. An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J Comput Appl Math, 2010, 235: 286–292

    Article  MATH  MathSciNet  Google Scholar 

  14. Ng M, Qi L, Zhou G L. Finding the largest eigenvalue of a non-negative tensor. SIAM J Matrx Anal Appl, 2009, 31: 1090–1099

    Article  MathSciNet  Google Scholar 

  15. Nussbaum R D. Hilbert’s Projective Metric and Iterated Nonlinear Maps. Boston, MA: Amer Math Soc, 1988

    Google Scholar 

  16. Pearson K J. Essentially positive tensors. Internat J Algebra, 2010, 4: 421–427

    MATH  Google Scholar 

  17. Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

    Article  MATH  MathSciNet  Google Scholar 

  18. Yang Q, Yang Y. Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236–1250

    Article  MATH  MathSciNet  Google Scholar 

  19. Yang Y, Yang Q. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517–2530

    Article  MATH  MathSciNet  Google Scholar 

  20. Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front Math China, 2011, 6: 363–378

    Article  MATH  MathSciNet  Google Scholar 

  21. Yang Y, Yang Q. Geometric simplicity of spectral radius of nonnegative irreducible tensors. Front Math China, 2013, 8: 129–140

    Article  MATH  MathSciNet  Google Scholar 

  22. Yang Y, Yang Q, Li Y. An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis. ArXiv:1102.2668, 2011

    Google Scholar 

  23. Zhang L, Qi L. Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor. Numer Linear Algebra Appl, 2012, 19: 830–841

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang L, Qi L, Luo L, et al. The dominant eigenvalue of an essentially nonnegative tensor. Numer Linear Algebra Appl, in press, 2013

    Google Scholar 

  25. Zhang L, Qi L, Xu Y. Linear convergence of the LZI algorithm for weakly positive tensors. J Comput Math, 2012, 30: 24–33

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang L, Qi L, Zhou G. M-tensors and the positive definiteness of a multivariate form. ArXiv:1202.6431, 2012

    Google Scholar 

  27. Zhou G L, Caccetta L, Qi L. Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor. Linear Algebra Appl, 2013, 438: 959–968

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    ShengLong Hu & LiQun Qi

  2. Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China

    ZhengHai Huang

  3. The Center for Applied Mathematics of Tianjin University, Tianjin University, Tianjin, 300072, China

    ZhengHai Huang

Authors
  1. ShengLong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. ZhengHai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. LiQun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ZhengHai Huang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, S., Huang, Z. & Qi, L. Strictly nonnegative tensors and nonnegative tensor partition. Sci. China Math. 57, 181–195 (2014). https://doi.org/10.1007/s11425-013-4752-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4752-4

Keywords

MSC(2010)

Search

Navigation