Short CommunicationC-eigenvalues intervals for piezoelectric-type tensors
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 be a third-order n dimensional real tensor. If the later two indices of are symmetric, i.e., for all j ∈ N and k ∈ N, where then 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 be a piezoelectric-type tensor. If there exists a scalar vectors and satisfying the following systemwhere and with the ith entryrespectively, then λ is called a C-eigenvalue of 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 . Thenwhere Furthermore, and are also C-eigenvalues and their associated C-eigenvectors of . Property 3 Suppose that λ* is the largest C-eigenvalue of a piezoelectric-type tensor . Then
Properties 2 and 3 provide theoretically the form to determine C-eigenvalues or the largest C-eigenvalue λ* of 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 be a piezoelectric-type tensor, and λ be a C-eigenvalue of . Thenwhere and . Proof Suppose that and are left and right C-eigenvectors corresponding to λ with and . Let
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 [2], [6], with entriesand zeroes elsewhere;
(II) The piezoelectric tensor [2], [4], [5], with entriesand zeroes elsewhere;
(III) The piezoelectric tensor [2], [6], with entries
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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