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Properties and calculation for \(C\)-eigenvalues of a piezoelectric-type tensor. (English) Zbl 1513.15024

Summary: This paper mainly considers the \(C\)-eigenvalues of a piezoelectric-type tensor. For this, we first discuss its relationship with \(l^{k, s}\)-singular values of a partially symmetric rectangular tensor, and then present three types of \(C\)-eigenvalue inclusion intervals which can be used to locate all \(C\)-eigenvalues of a piezoelectric-type tensor and can provide an upper and a lower bound for the largest \(C\)-eigenvalue of a piezoelectric-type tensor. Finally, we present an alternative method to compute all \(C\)-eigenpairs of a piezoelectric-type tensor.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
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[1] L. V. Ahlfors, Complex Analysis, 2nd edn, McGraw-Hill, New York, 1966. · Zbl 0142.01701
[2] K. Chang; L. Qi; G. Zhou, Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370, 284-294 (2010) · Zbl 1201.15003
[3] H. Che; H. Chen; Y. Wang, \(C\)-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett., 89, 41-49 (2019) · Zbl 1444.15013
[4] Y. Chen, A. Jákli and L. Qi, Spectral analysis of piezoelectric tensors, preprint, arXiv: 1703.07937v1.
[5] Y. Chen, L. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20 pp. · Zbl 1383.82065
[6] Z. Chen; L. Qi; Q. Yang; Y. Yang, The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis, Linear Algebra Appl., 439, 3713-3733 (2013) · Zbl 1283.65058
[7] J. Curie; P. Curie, Développement, par compression de l’électricité polaire dans les cristaux hémiédres à faces inclinées, Bulletin de Minéralogie, 3-4, 90-93 (1880)
[8] G. Dahl; J. M. Leinaas; J. Myrheim; E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420, 711-725 (2007) · Zbl 1118.15027
[9] M. De Jong; W. Chen; H. Geerlings; M. Asta; K. A. Persson, A database to enable discovery and design of piezoelectric materials, Sci. Data, 2, 150053 (2015)
[10] W. Ding; Z. Hou; Y. Wei, Tensor logarithmic norm and its applications, Numer. Linear Algebra Appl., 23, 989-1006 (2016) · Zbl 1424.15046
[11] A. Einstein; B. Podolsky; N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys Rev., 47, 777-780 (1935) · Zbl 0012.04201
[12] G. H. Golub and C. F. Van Loan, Matrix Computations (4th edn), Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press: Baltimore, MD, 2013. · Zbl 1268.65037
[13] J. He, Y. Liu, G. Xu and G. Liu, \(V\)-singular values of rectangular tensors and their applications, J. Inequal. Appl., 2019 (2019), Paper No. 84, 15 pp. · Zbl 1499.15029
[14] A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008.
[15] J. K. Knowles; E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elast., 5, 341-361 (1975) · Zbl 0323.73010
[16] C. Li; Y. Liu; Y. Li, \(C\)-eigenvalues intervals for piezoelectric-type tensors, Appl. Math. Comput., 358, 244-250 (2019) · Zbl 1428.15025
[17] S. Li, Z. Chen, C. Li and J. Zhao, Eigenvalue bounds of third-order tensors via the minimax eigenvalue of symmetric matrices, Comput. Appl. Math., 39 (2020), Paper No. 217, 14 pp. · Zbl 1463.15025
[18] W. Li; R. Ke; W.-K. Ching; M. K. Ng, A \(C\)-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains, Comput. Math. Appl., 78, 1008-1025 (2019) · Zbl 1442.15043
[19] L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, (2005), 129-132.
[20] C. Ling; L. Qi, \(l^{k, s} \)-Singular values and spectral radius of rectangular tensors, Front. Math. China, 8, 63-83 (2013) · Zbl 1272.15008
[21] X. Liu, S. Yin and H. Li, \(C\)-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices, J. Ind. Manag. Optim., (2020). · Zbl 1476.15016
[22] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40, 1302-1324 (2005) · Zbl 1125.15014
[23] L. Qi, Transposes, \(L\)-eigenvalues and invariants of third order tensors, preprint, (2017), arXiv: 1704.01327.
[24] L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, Springer, Singapore, 2018. · Zbl 1398.15001
[25] L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. · Zbl 1370.15001
[26] C. Sang, An \(S\)-type singular value inclusion set for rectangular tensors, J. Inequal. Appl., 2017 (2017), Paper No. 141, 14 pp. · Zbl 1365.15012
[27] L. Sun; G. Wang; L. Liu, Further study on \(Z\)-eigenvalue localization set and positive definiteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 44, 105-129 (2021) · Zbl 1458.15022
[28] W. Wang; H. Chen; Y. Wang, A new \(C\)-eigenvalue interval for piezoelectric-type tensors, Appl. Math. Lett., 100, 106035 (2020) · Zbl 1524.15028
[29] Y. Wang; M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J Elast., 44, 89-96 (1996) · Zbl 0876.73030
[30] L. Xiong; J. Liu, A new \(C\)-eigenvalue localisation set for piezoelectric-type tensors, E. Asian J. Appl. Math., 10, 123-134 (2020) · Zbl 1464.15019
[31] Y. Yang; Q. Yang, Singular values of nonnegative rectangular tensors, Front. Math. China, 6, 363-378 (2011) · Zbl 1213.74060
[32] H. Yao; B. Long; C. Bu; J. Zhou, \(l^{k, s} \)-Singular values and spectral radius of partially symmetric rectangular tensors, Front. Math. China, 11, 605-622 (2016) · Zbl 1362.15021
[33] J. Zhao, Two new singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 67, 2451-2470 (2019) · Zbl 1423.15027
[34] J. Zhao; C. Li, Singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 66, 1333-1350 (2018) · Zbl 1391.15088
[35] W.-N. Zou; C.-X. Tang; E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469, 20120755 (2013)
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