M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors

Haitao Che,Yiju Wang,Haibin Chen

doi:10.1186/s13660-019-1986-x

Haitao Che, Yiju Wang + Show 1 more

Open Access

https://doi.org/10.1186/s13660-019-1986-x

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Abstract

Inspired by symmetric Cauchy tensors, we define fourth-order partially symmetric Cauchy tensors with their generating vectors. In this article, we focus on the necessary and sufficient conditions for the M-positive semi-definiteness and M-positive definiteness of fourth-order Cauchy tensors. Moreover, the necessary and sufficient conditions of the strong ellipticity conditions for fourth-order Cauchy tensors are obtained. Furthermore, fourth-order Cauchy tensors are M-positive semi-definite if and only if the homogeneous polynomial for fourth-order Cauchy tensors is monotonically increasing. Several M-eigenvalue inclusion theorems and spectral properties of fourth-order Cauchy tensors are discussed. A power method is proposed to compute the smallest and the largest M-eigenvalues of fourth-order Cauchy tensors. The given numerical experiments show the effectiveness of the proposed method.

Highlights

Let Rn be an n-dimensional real Euclidean space and denote the set consisting of all natural numbers by N
The nonlinear elastic materials analysis and entanglement studies in quantum physics can be formulated as the following optimization problem:
In the nonlinear elastic materials analysis, one approach is to consider an elastic material in terms of a fourth-order three-dimensional elastic module tensor that satisfies the partially symmetric property [1]

Summary

Introduction

Let Rn be an n-dimensional real Euclidean space and denote the set consisting of all natural numbers by N. Several spectral properties of M-positive semi-definite fourth-order Cauchy tensors are discussed. We will show some necessary and sufficient conditions for fourth-order Cauchy tensors to be M-positive semi-definite.

Results

Conclusion