Elsevier

Applied Mathematics and Computation

Volume 358, 1 October 2019, Pages 244-250
Applied Mathematics and Computation

Short Communication
C-eigenvalues intervals for piezoelectric-type tensors

https://doi.org/10.1016/j.amc.2019.04.036Get rights and content

Abstract

C-eigenvalues of piezoelectric-type tensors are real and always exist, and the largest C-eigenvalue for the piezoelectric tensor determines the highest piezoelectric coupling constant. In this paper, we give two intervals to locate all C-eigenvalues for a given piezoelectric-type tensor. These intervals provide upper bounds for the largest C-eigenvalue. Numerical examples are also given to show the corresponding results.

Introduction

Piezoelectric-type tensors are introduced by Chen et al. in [2] as a subclass of third order tensors which have extensive applications in physics and engineering [3], [4], [5], [7], [8], [9], [10], [11], [14]. The class of piezoelectric tensors, as the subclass of Piezoelectric-type tensors of dimension three, plays the key role in Piezoelectric effect and converse Piezoelectric effect [2].

Definition 1

[2, Definition 2.1] Let A=(aijk)Rn×n×n be a third-order n dimensional real tensor. If the later two indices of A are symmetric, i.e., aijk=aikj for all j ∈ N and k ∈ N, where N:={1,2,n}, then A is called a piezoelectric-type tensor.

To explore more properties related to piezoelectric effect and converse piezoelectric effect in solid crystal, Chen et al. in [2] introduced C-eigenvalues and C-eigenvectors for Piezoelectric-type tensors, and shown that the largest C-eigenvalue corresponds to the electric displacement vector with the largest 2-norm in the piezoelectric electronic effect under unit uniaxial stress [2], [4], [13].

Definition 2

[2, Definition 2.2] Let A=(aijk)Rn×n×n be a piezoelectric-type tensor. If there exists a scalar λR, vectors xRn and yRn satisfying the following systemAyy=λx,xAy=λy,xTx=1andyTy=1,where AyyRn and xAyRn with the ith entry(Ayy)i=j,kNaijkyjyk,and(xAy)i=j,kNajkixjyk,respectively, then λ is called a C-eigenvalue of A, and x, y are called associated left and right C-eigenvectors, respectively.

For C-eigenvalues and associated left and right C-eigenvectors of a piezoelectric-type tensor, Chen et al. in [2] also provided several related results, such as:

Property 1

Any piezoelectric-type tensor always has, at least, one C-eigenvalue, with associated right and left C-eigenvectors.

Property 2

Suppose that λ, and x, y are a C-eigenvalue, and its associated left and right C-eigenvectors of a piezoelectric-type tensor A. Thenλ=xAyy,where xAyy=i,j,kNaijkxiyjyk. Furthermore, (λ,x,y), (λ,x,y) and (λ,x,y) are also C-eigenvalues and their associated C-eigenvectors of A.

Property 3

Suppose that λ* is the largest C-eigenvalue of a piezoelectric-type tensor A. Thenλ*=max{xAyy:xTx=1,yTy=1}.

Properties 2 and 3 provide theoretically the form to determine C-eigenvalues or the largest C-eigenvalue λ* of A, However, it is difficult to compute them in practice because determining x and y is not easy. So, we in this paper give two intervals to locate all C-eigenvalues of a piezoelectric-type tensor, and then give some upper bounds for the the largest C-eigenvalue. This can provide more information before calculating them out.

Section snippets

Main results

In this section, we give two intervals to locate all C-eigenvalues of a piezoelectric-type tensor. And the comparison of these two intervals are also established.

Theorem 1

Let A=(aijk)Rn×n×n be a piezoelectric-type tensor, and λ be a C-eigenvalue of A. Thenλ[ρ,ρ],whereρ:=maxi,jN(Ri(1)(A)Rj(3)(A))12,Ri(1)(A):=l,kN|ailk| and Rj(3)(A):=l,kN|alkj|.

Proof

Suppose that x=(x1,x2,,xn)T and y=(y1,y2,,yn)T are left and right C-eigenvectors corresponding to λ with xTx=1 and yTy=1. Let|xp|=maxiN|xi|,and|yq|=maxi

Numerical examples

In this section, we give some examples to show the results obtained above. Consider the eight piezoelectric tensors in [2];

(I) The piezoelectric tensor AVFeSb [2], [6], with entriesa123=a213=a312=3.68180677,and zeroes elsewhere;

(II) The piezoelectric tensor ASiO2 [2], [4], [5], with entriesa111=a122=a212=0.13685,anda123=a213=0.009715,and zeroes elsewhere;

(III) The piezoelectric tensor ACr2AgBiO8 [2], [6], with entriesa123=a213=0.22163,a113=a223=2.608665,a311=a322=0.152485,anda312=

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions and comments, and handling editor for suggestions on languages.

Chaoqian Li’s work is supported in part by the Applied Basic Research Programs of Science and Technology Department of Yunnan Province (2018FB001); Program for Excellent Young Talents, Yunnan University; Outstanding Youth Cultivation Project for Yunnan Province (2018YDJQ021); Yunnan Provincial Ten Thousands Plan Young Top Talents; Shanghai Key

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