doi: 10.3934/jimo.2021162
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Properties and calculation for C-eigenvalues of a piezoelectric-type tensor

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

* Corresponding author: Jianxing Zhao

Received  March 2021 Revised  June 2021 Early access September 2021

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.

Citation: Jianxing Zhao, Jincheng Luo. Properties and calculation for C-eigenvalues of a piezoelectric-type tensor. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021162
References:
[1]

L. V. Ahlfors, Complex Analysis, 2nd edn, McGraw-Hill, New York, 1966.

[2]

K. ChangL. Qi and G. Zhou, Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294.  doi: 10.1016/j.jmaa.2010.04.037.

[3]

H. CheH. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett., 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.

[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. doi: 10.1088/1751-8121/aa98a8.

[6]

Z. ChenL. QiQ. Yang and Y. Yang, The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis, Linear Algebra Appl., 439 (2013), 3713-3733.  doi: 10.1016/j.laa.2013.09.027.

[7]

J. Curie and P. Curie, Développement, par compression de l'électricité polaire dans les cristaux hémiédres à faces inclinées, Bulletin de Minéralogie, 3-4 (1880), 90-93.  doi: 10.3406/bulmi.1880.1564.

[8]

G. DahlJ. M. LeinaasJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

[9]

M. De JongW. ChenH. GeerlingsM. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Sci. Data, 2 (2015), 150053.  doi: 10.1038/sdata.2015.53.

[10]

W. DingZ. Hou and Y. Wei, Tensor logarithmic norm and its applications, Numer. Linear Algebra Appl., 23 (2016), 989-1006.  doi: 10.1002/nla.2064.

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A. EinsteinB. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys Rev., 47 (1935), 777-780.  doi: 10.1103/PhysRev.47.777.

[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.

[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. doi: 10.1186/s13660-019-2036-4.

[14]

A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008.

[15]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elast., 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[16]

C. LiY. Liu and Y. Li, C-eigenvalues intervals for piezoelectric-type tensors, Appl. Math. Comput., 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.

[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. doi: 10.1007/s40314-020-01245-0.

[18]

W. LiR. KeW.-K. Ching and M. K. Ng, A $C$-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains, Comput. Math. Appl., 78 (2019), 1008-1025.  doi: 10.1016/j.camwa.2019.03.016.

[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 and L. Qi, lk, s-Singular values and spectral radius of rectangular tensors, Front. Math. China, 8 (2013), 63-83.  doi: 10.1007/s11464-012-0265-7.

[21]

X. Liu, S. Yin and H. Li, C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices, J. Ind. Manag. Optim., (2020). doi: 10.3934/jimo.2020122.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[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. doi: 10.1007/978-981-10-8058-6.

[25]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.

[26]

C. Sang, An S-type singular value inclusion set for rectangular tensors, J. Inequal. Appl., 2017 (2017), Paper No. 141, 14 pp. doi: 10.1186/s13660-017-1421-0.

[27]

L. SunG. Wang and L. Liu, Further study on Z-eigenvalue localization set and positive definiteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 44 (2021), 105-129.  doi: 10.1007/s40840-020-00939-2.

[28]

W. WangH. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Appl. Math. Lett., 100 (2020), 106035.  doi: 10.1016/j.aml.2019.106035.

[29]

Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J Elast., 44 (1996), 89-96.  doi: 10.1007/BF00042193.

[30]

L. Xiong and J. Liu, A new C-eigenvalue localisation set for piezoelectric-type tensors, E. Asian J. Appl. Math., 10 (2020), 123-134.  doi: 10.4208/eajam.060119.040619.

[31]

Y. Yang and Q. Yang, Singular values of nonnegative rectangular tensors, Front. Math. China, 6 (2011), 363-378.  doi: 10.1007/s11464-011-0108-y.

[32]

H. YaoB. LongC. Bu and J. Zhou, lk, s-Singular values and spectral radius of partially symmetric rectangular tensors, Front. Math. China, 11 (2016), 605-622.  doi: 10.1007/s11464-015-0494-7.

[33]

J. Zhao, Two new singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 67 (2019), 2451-2470.  doi: 10.1080/03081087.2018.1494125.

[34]

J. Zhao and C. Li, Singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 66 (2018), 1333-1350.  doi: 10.1080/03081087.2017.1351518.

[35]

W.-N. ZouC.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755.  doi: 10.1098/rspa.2012.0755.

show all references

References:
[1]

L. V. Ahlfors, Complex Analysis, 2nd edn, McGraw-Hill, New York, 1966.

[2]

K. ChangL. Qi and G. Zhou, Singular values of a real rectangular tensor, J. Math. Anal. Appl., 370 (2010), 284-294.  doi: 10.1016/j.jmaa.2010.04.037.

[3]

H. CheH. Chen and Y. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett., 89 (2019), 41-49.  doi: 10.1016/j.aml.2018.09.014.

[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. doi: 10.1088/1751-8121/aa98a8.

[6]

Z. ChenL. QiQ. Yang and Y. Yang, The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis, Linear Algebra Appl., 439 (2013), 3713-3733.  doi: 10.1016/j.laa.2013.09.027.

[7]

J. Curie and P. Curie, Développement, par compression de l'électricité polaire dans les cristaux hémiédres à faces inclinées, Bulletin de Minéralogie, 3-4 (1880), 90-93.  doi: 10.3406/bulmi.1880.1564.

[8]

G. DahlJ. M. LeinaasJ. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725.  doi: 10.1016/j.laa.2006.08.026.

[9]

M. De JongW. ChenH. GeerlingsM. Asta and K. A. Persson, A database to enable discovery and design of piezoelectric materials, Sci. Data, 2 (2015), 150053.  doi: 10.1038/sdata.2015.53.

[10]

W. DingZ. Hou and Y. Wei, Tensor logarithmic norm and its applications, Numer. Linear Algebra Appl., 23 (2016), 989-1006.  doi: 10.1002/nla.2064.

[11]

A. EinsteinB. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys Rev., 47 (1935), 777-780.  doi: 10.1103/PhysRev.47.777.

[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.

[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. doi: 10.1186/s13660-019-2036-4.

[14]

A. Kholkin, N. Pertsev and A. Goltsev, Piezolelectricity and Crystal Symmetry, Piezoelectric and Acoustic Materials, Springer, New York, 2008.

[15]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elast., 5 (1975), 341-361.  doi: 10.1007/BF00126996.

[16]

C. LiY. Liu and Y. Li, C-eigenvalues intervals for piezoelectric-type tensors, Appl. Math. Comput., 358 (2019), 244-250.  doi: 10.1016/j.amc.2019.04.036.

[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. doi: 10.1007/s40314-020-01245-0.

[18]

W. LiR. KeW.-K. Ching and M. K. Ng, A $C$-eigenvalue problem for tensors with applications to higher-order multivariate Markov chains, Comput. Math. Appl., 78 (2019), 1008-1025.  doi: 10.1016/j.camwa.2019.03.016.

[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 and L. Qi, lk, s-Singular values and spectral radius of rectangular tensors, Front. Math. China, 8 (2013), 63-83.  doi: 10.1007/s11464-012-0265-7.

[21]

X. Liu, S. Yin and H. Li, C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices, J. Ind. Manag. Optim., (2020). doi: 10.3934/jimo.2020122.

[22]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.

[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. doi: 10.1007/978-981-10-8058-6.

[25]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.

[26]

C. Sang, An S-type singular value inclusion set for rectangular tensors, J. Inequal. Appl., 2017 (2017), Paper No. 141, 14 pp. doi: 10.1186/s13660-017-1421-0.

[27]

L. SunG. Wang and L. Liu, Further study on Z-eigenvalue localization set and positive definiteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 44 (2021), 105-129.  doi: 10.1007/s40840-020-00939-2.

[28]

W. WangH. Chen and Y. Wang, A new C-eigenvalue interval for piezoelectric-type tensors, Appl. Math. Lett., 100 (2020), 106035.  doi: 10.1016/j.aml.2019.106035.

[29]

Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J Elast., 44 (1996), 89-96.  doi: 10.1007/BF00042193.

[30]

L. Xiong and J. Liu, A new C-eigenvalue localisation set for piezoelectric-type tensors, E. Asian J. Appl. Math., 10 (2020), 123-134.  doi: 10.4208/eajam.060119.040619.

[31]

Y. Yang and Q. Yang, Singular values of nonnegative rectangular tensors, Front. Math. China, 6 (2011), 363-378.  doi: 10.1007/s11464-011-0108-y.

[32]

H. YaoB. LongC. Bu and J. Zhou, lk, s-Singular values and spectral radius of partially symmetric rectangular tensors, Front. Math. China, 11 (2016), 605-622.  doi: 10.1007/s11464-015-0494-7.

[33]

J. Zhao, Two new singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 67 (2019), 2451-2470.  doi: 10.1080/03081087.2018.1494125.

[34]

J. Zhao and C. Li, Singular value inclusion sets for rectangular tensors, Linear Multilinear Algebra, 66 (2018), 1333-1350.  doi: 10.1080/03081087.2017.1351518.

[35]

W.-N. ZouC.-X. Tang and E. Pan, Symmetry types of the piezoelectric tensor and their identification, Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 469 (2013), 20120755.  doi: 10.1098/rspa.2012.0755.

Table 1.  Numerical comparison among $\varrho$, $\varrho_{\min}$, $\tilde{\varrho}_{\min}$, $\rho_{\Gamma}$, $\rho_{\mathcal{L}}$, $\rho_{\mathcal{M}}$, $\rho_{\Upsilon}$, $\rho_{\gamma}$, $\rho_C$, $\rho_G$, $\rho_B$, $\rho_{\min}$ and $\lambda^*$.
$\mathcal{A}_{\rm{VFeSb}}$ $\mathcal{A}_{\rm SiO_2}$ $\mathcal{A}_{\rm Cr_2AgBiO_8}$ $\mathcal{A}_{\rm RbTaO_3}$ $\mathcal{A}_{\rm NaBiS_2}$ $\mathcal{A}_{\rm LiBiB_2O_5}$ $\mathcal{A}_{\rm KBi_2F_7}$ $\mathcal{A}_{\rm BaNiO_3}$
$\varrho$ 7.3636 0.2882 5.6606 30.0911 17.3288 15.2911 22.6896 33.7085
$\varrho_{\min}$ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$\tilde{\varrho}_{\min}$ 7.3636 0.2393 4.6717 22.7163 14.5723 12.1694 18.7025 27.5396
$\rho_{\Gamma}$ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$\rho_\mathcal{L}$ 7.3636 0.2744 4.8058 23.5377 16.5640 11.0127 18.8793 27.5109
$\rho_\mathcal{M}$ 7.3636 0.2834 4.7861 23.5377 16.8464 11.0038 19.8830 27.5013
$\rho_{\Upsilon}$ 7.3636 0.2834 4.7335 23.5377 16.8464 10.9998 19.8319 27.5013
$\rho_{\gamma}$ 7.3636 0.2744 4.7860 23.0353 16.4488 10.2581 18.4090 27.5013
$\rho_C$ 6.3771 0.1943 3.7242 16.0259 11.9319 7.7540 13.5113 27.4629
$\rho_G$ 6.3771 0.2506 4.0455 21.5313 13.9063 9.8718 14.2574 29.1441
$\rho_B$ 5.2069 0.2345 4.0026 19.4558 13.4158 10.0289 15.3869 27.5396
$\rho_{\min}$ 6.5906 0.1942 3.5097 18.0991 11.9324 8.1373 14.3299 27.4725
$\lambda^*$ 4.2514 0.1375 2.6258 12.4234 11.6674 7.7376 13.5021 27.4628
$\mathcal{A}_{\rm{VFeSb}}$ $\mathcal{A}_{\rm SiO_2}$ $\mathcal{A}_{\rm Cr_2AgBiO_8}$ $\mathcal{A}_{\rm RbTaO_3}$ $\mathcal{A}_{\rm NaBiS_2}$ $\mathcal{A}_{\rm LiBiB_2O_5}$ $\mathcal{A}_{\rm KBi_2F_7}$ $\mathcal{A}_{\rm BaNiO_3}$
$\varrho$ 7.3636 0.2882 5.6606 30.0911 17.3288 15.2911 22.6896 33.7085
$\varrho_{\min}$ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$\tilde{\varrho}_{\min}$ 7.3636 0.2393 4.6717 22.7163 14.5723 12.1694 18.7025 27.5396
$\rho_{\Gamma}$ 7.3636 0.2834 5.6606 23.5377 16.8548 12.3206 20.2351 27.5396
$\rho_\mathcal{L}$ 7.3636 0.2744 4.8058 23.5377 16.5640 11.0127 18.8793 27.5109
$\rho_\mathcal{M}$ 7.3636 0.2834 4.7861 23.5377 16.8464 11.0038 19.8830 27.5013
$\rho_{\Upsilon}$ 7.3636 0.2834 4.7335 23.5377 16.8464 10.9998 19.8319 27.5013
$\rho_{\gamma}$ 7.3636 0.2744 4.7860 23.0353 16.4488 10.2581 18.4090 27.5013
$\rho_C$ 6.3771 0.1943 3.7242 16.0259 11.9319 7.7540 13.5113 27.4629
$\rho_G$ 6.3771 0.2506 4.0455 21.5313 13.9063 9.8718 14.2574 29.1441
$\rho_B$ 5.2069 0.2345 4.0026 19.4558 13.4158 10.0289 15.3869 27.5396
$\rho_{\min}$ 6.5906 0.1942 3.5097 18.0991 11.9324 8.1373 14.3299 27.4725
$\lambda^*$ 4.2514 0.1375 2.6258 12.4234 11.6674 7.7376 13.5021 27.4628
Table 2.  Z-eigenvalues $ \lambda_z $ and their parts of Z-eigenvectors $ z = (z_1, \ldots, z_6)^\top $ of $ \mathcal{C}_{\mathcal{A}_{\rm{VFeSb}}} $.
$ \lambda_z $ $ z_1 $ $ z_2 $ $ z_3 $ $ z_4 $ $ z_5 $ $ z_6 $
$ -3.0062 $ 0.4082 0.4082 0.4082 0.4082 0.4082 0.4082
$ -2.6034 $ 0.7071 0 0 0 0.5000 0.5000
0 0.0077 0 0 1.0000 0 0
2.6034 $ -0.7071 $ 0 0 0 $ -0.5000 $ $ -0.5000 $
3.0062 $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $
$ \lambda_z $ $ z_1 $ $ z_2 $ $ z_3 $ $ z_4 $ $ z_5 $ $ z_6 $
$ -3.0062 $ 0.4082 0.4082 0.4082 0.4082 0.4082 0.4082
$ -2.6034 $ 0.7071 0 0 0 0.5000 0.5000
0 0.0077 0 0 1.0000 0 0
2.6034 $ -0.7071 $ 0 0 0 $ -0.5000 $ $ -0.5000 $
3.0062 $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $ $ -0.4082 $
Table 3.  All C-eigenvalues $ \lambda $ and their parts of C-eigenvectors pairs $ (x, y) $ of $ \mathcal{A} $.
$ \lambda $ $ x_1 $ $ x_2 $ $ x_3 $ $ y_1 $ $ y_2 $ $ y_3 $
$ -4.2514 $ 0.5774 0.5774 0.5774 0.5774 0.5774 0.5774
$ -3.6818 $ 1.0000 0 0 0 0.7071 0.7071
0 1.0000 0 0 1.0000 0 0
3.6818 $ -1.0000 $ 0 0 0 $ -0.7071 $ $ -0.7071 $
4.2514 $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $
$ \lambda $ $ x_1 $ $ x_2 $ $ x_3 $ $ y_1 $ $ y_2 $ $ y_3 $
$ -4.2514 $ 0.5774 0.5774 0.5774 0.5774 0.5774 0.5774
$ -3.6818 $ 1.0000 0 0 0 0.7071 0.7071
0 1.0000 0 0 1.0000 0 0
3.6818 $ -1.0000 $ 0 0 0 $ -0.7071 $ $ -0.7071 $
4.2514 $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $ $ -0.5774 $
Table 4.  Z-eigenvalues $ \lambda_z $ and their parts of Z-eigenvectors $ z = (z_1, \ldots, z_6)^\top $ of $ \mathcal{C}_{\mathcal{A}} $.
$ \lambda_z $ $ z_1 $ $ z_2 $ $ z_3 $ $ z_4 $ $ z_5 $ $ z_6 $
$ -12.7279 $ 0 0 $ -0.7071 $ 0 0 0.7071
$ -7.8102 $ $ -0.2561 $ $ -0.3201 $ $ -0.5762 $ 0.5000 0.5000 0
$ -5.0000 $ 0.4000 0.5000 $ -0.3000 $ $ -0.5000 $ 0.5000 0
$ -3.8419 $ 0.1874 0.2343 $ -0.6403 $ $ -0.6952 $ 0.1295 0
0 1.0000 $ -0.0037 $ 0.0000 0 0 0
3.8419 $ -0.1874 $ $ -0.2343 $ 0.6403 0.6952 $ -0.1295 $ 0
5.0000 $ -0.4000 $ $ -0.5000 $ 0.3000 0.5000 $ -0.5000 $ 0
7.8102 0.2561 0.3201 0.5762 0.5000 0.5000 0
12.7279 0 0 0.7071 0 0 0.7071
$ \lambda_z $ $ z_1 $ $ z_2 $ $ z_3 $ $ z_4 $ $ z_5 $ $ z_6 $
$ -12.7279 $ 0 0 $ -0.7071 $ 0 0 0.7071
$ -7.8102 $ $ -0.2561 $ $ -0.3201 $ $ -0.5762 $ 0.5000 0.5000 0
$ -5.0000 $ 0.4000 0.5000 $ -0.3000 $ $ -0.5000 $ 0.5000 0
$ -3.8419 $ 0.1874 0.2343 $ -0.6403 $ $ -0.6952 $ 0.1295 0
0 1.0000 $ -0.0037 $ 0.0000 0 0 0
3.8419 $ -0.1874 $ $ -0.2343 $ 0.6403 0.6952 $ -0.1295 $ 0
5.0000 $ -0.4000 $ $ -0.5000 $ 0.3000 0.5000 $ -0.5000 $ 0
7.8102 0.2561 0.3201 0.5762 0.5000 0.5000 0
12.7279 0 0 0.7071 0 0 0.7071
Table 5.  All C-eigenvalues $ \lambda $ and their parts of C-eigenvectors pairs $ (x, y) $ of $ \mathcal{A} $.
$ \lambda $ $ x_1 $ $ x_2 $ $ x_3 $ $ y_1 $ $ y_2 $ $ y_3 $
$ -18.0000 $ 0 0 $ -1.0000 $ 0 0 1.0000
$ -11.0454 $ $ -0.3621 $ $ -0.4527 $ $ -0.8148 $ 0.7071 0.7071 0
$ -7.0711 $ 0.5657 0.7071 $ -0.4243 $ $ -0.7071 $ 0.7071 0
$ -5.4332 $ 0.2650 0.3313 $ -0.9055 $ $ -0.9831 $ 0.1831 0
5.4332 $ -0.2650 $ $ -0.3313 $ 0.9055 0.9831 $ -0.1831 $ 0
7.0711 $ -0.5657 $ $ -0.7071 $ 0.4243 0.7071 $ -0.7071 $ 0
11.0454 0.3621 0.4527 0.8148 0.7071 0.7071 0
18.0000 0 0 1.0000 0 0 1.0000
$ \lambda $ $ x_1 $ $ x_2 $ $ x_3 $ $ y_1 $ $ y_2 $ $ y_3 $
$ -18.0000 $ 0 0 $ -1.0000 $ 0 0 1.0000
$ -11.0454 $ $ -0.3621 $ $ -0.4527 $ $ -0.8148 $ 0.7071 0.7071 0
$ -7.0711 $ 0.5657 0.7071 $ -0.4243 $ $ -0.7071 $ 0.7071 0
$ -5.4332 $ 0.2650 0.3313 $ -0.9055 $ $ -0.9831 $ 0.1831 0
5.4332 $ -0.2650 $ $ -0.3313 $ 0.9055 0.9831 $ -0.1831 $ 0
7.0711 $ -0.5657 $ $ -0.7071 $ 0.4243 0.7071 $ -0.7071 $ 0
11.0454 0.3621 0.4527 0.8148 0.7071 0.7071 0
18.0000 0 0 1.0000 0 0 1.0000
[1]

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Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure and Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

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