Bi-block positive semidefiniteness of bi-block symmetric tensors

doi:10.1007/s11464-021-0874-0

Front. Math. China

2021, Vol. 16

Issue (1) : 141-169 https://doi.org/10.1007/s11464-021-0874-0

RESEARCH ARTICLE

Bi-block positive semidefiniteness of bi-block symmetric tensors

Zheng-Hai HUANG, Xia LI, Yong WANG(

)

School of Mathematics, Tianjin University, Tianjin 300354, China

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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 suffcient conditions for the strong ellipticity of elasticity tensors.

Keywords Bi-block symmetric tensor bi-block symmetric Z-tensor bi-block symmetric B₀-tensor diagonally dominant bi-block symmetric tensor bi-block M-eigenvalue

Corresponding Author(s): Yong WANG

Issue Date: 26 March 2021

Cite this article:

Zheng-Hai HUANG,Xia LI,Yong WANG. Bi-block positive semidefiniteness of bi-block symmetric tensors[J]. Front. Math. China, 2021, 16(1): 141-169.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0874-0
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I1/141

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