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Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors


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.

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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).

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Correspondence to Xinzhen Zhang.

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Qi, L., Hu, S., Zhang, X. et al. Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors. Front. Math. China 16, 171–185 (2021).

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


  • 15A69