Skip to main content

Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors

Abstract

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Comon P, Golub G, Lim L, Mourrain B. Symmetric tensors and symmetric tensor rank. SIAM J Matrix Anal Appl, 2008, 30(3): 1254–1279

    MathSciNet  Article  Google Scholar 

  2. 2.

    De Lathauwer L, De Moor B, Vandewalle J. A multilinear singular value decomposition. SIAM J Matrix Anal Appl, 2000, 21(4): 1253–1278

    MathSciNet  Article  Google Scholar 

  3. 3.

    Friedland S, Lim L. Nuclear norm of high-order tensors. Math Comp, 2018, 87(311): 1255–1281

    MathSciNet  Article  Google Scholar 

  4. 4.

    Hu S. Relations of the nuclear norm of a tensor and its matrix flattenings. Linear Algebra Appl, 2015, 478: 188–199

    MathSciNet  Article  Google Scholar 

  5. 5.

    Jiang B, Yang F, Zhang S. Tensor and its tucker core: The invariance relationships. Numer Linear Algebra Appl, 2017, 24(3): e2086

    MathSciNet  Article  Google Scholar 

  6. 6.

    Knowles J K, Sternberg E. On the ellipticity of the equations of nonlinear elastostatistics for a special material. J. Elasticity, 1975, 5: 341–361

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kolda T, Bader B. Tensor decomposition and applications. SIAM Rev, 2009, 51(3): 455–500

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ling C, Nie J, Qi L, Ye Y. Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J Matrix Anal Appl, 2009, 20(3): 1286–1310

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Nye J F. Physical Properties of Crystals: Their Representation by Tensors and Matrices. 2nd ed. Oxford: Clarendon Press, 1985

  10. 10.

    Qi L, Dai H, Han D. Conditions for strong ellipticity and M-eigenvalues. Front Math China, 2009, 4(2): 349–364

    MathSciNet  Article  Google Scholar 

  11. 11.

    Qi L, Hu S, Xu Y. Spectral norm and nuclear norm of a third order tensor. J Ind Manag Optim, https://doi.org/10.3934/jimo.2021010

  12. 12.

    Qi L, Hu S, Zhang X, Chen Y. Tensor norm, cubic power and Gelfand limit. arXiv: 1909.10942

  13. 13.

    Rindler W. Relativity: Special, General and Cosmological. 2nd ed. Oxford: Oxford Univ Press, 2006

  14. 14.

    Simpson H, Spector S. On copositive matrices and strong ellipticity for isotropic elastic materials. Arch Ration Mech Anal, 1983, 84: 55–68

    MathSciNet  Article  Google Scholar 

  15. 15.

    Wang Y, Qi L, Zhang X. A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer Linear Algebra Appl, 2009, 16(7): 589–601

    MathSciNet  Article  Google Scholar 

  16. 16.

    Xiang H, Qi L, Wei Y. M-eigenvalues of Riemann curvature tensor. Commun Math Sci, 2018, 16(8): 2301–2315

    MathSciNet  Article  Google Scholar 

  17. 17.

    Yuan M, Zhang C. On tensor completion via nuclear norm minimization. Found Comput Math, 2016, 16(4): 1031–1068

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zou W, He Q, Huang M, Zheng Q. Eshelby’s problem of non-elliptical inclusions. J Mech Phys Solids, 2010, 58(3): 346–372

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Bo Jiang for the discussion on Theorem 6, and to Chen Ling for his comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771328, 11871369) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LD19A010002).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xinzhen Zhang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qi, L., Hu, S., Zhang, X. et al. Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors. Front. Math. China 16, 171–185 (2021). https://doi.org/10.1007/s11464-021-0895-8

Download citation

Keywords

  • Biquadratic tensor
  • nuclear norm
  • tensor product
  • biquadratic rank-one decomposition
  • biquadratic Tucker decomposition

MSC2020

  • 15A69