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Bi-block positive semidefiniteness of bi-block symmetric tensors

Abstract

The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.

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Acknowledgements

The first author’s work was supported by the National Natural Science Foundation of China (Grant No. 11871051).

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Correspondence to Yong Wang.

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Huang, ZH., Li, X. & Wang, Y. Bi-block positive semidefiniteness of bi-block symmetric tensors. Front. Math. China 16, 141–169 (2021). https://doi.org/10.1007/s11464-021-0874-0

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Keywords

  • Bi-block symmetric tensor
  • bi-block symmetric Z-tensor
  • bi-block symmetric B 0-tensor
  • diagonally dominant bi-block symmetric tensor
  • bi-block M-eigenvalue

MSC2020

  • 15A18
  • 15A69
  • 15B99