February  2022, 5(1): 33-44. doi: 10.3934/mfc.2021018

Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China

*Corresponding author: Gang Wang

Received  June 2021 Revised  August 2021 Published  February 2022 Early access  September 2021

Fund Project: The authors were supported by the Natural Science Foundation of Shandong Province (ZR2020MA025), the Natural Science Foundation of China (12071250) and High Quality Curriculum of Postgraduate Education in Shandong Province (SDYKC20109)

$ M $-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum $ M $-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.

Citation: Yuyan Yao, Gang Wang. Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors. Mathematical Foundations of Computing, 2022, 5 (1) : 33-44. doi: 10.3934/mfc.2021018
References:
[1]

H. CheH. Chen and Y. Wang, On the $M$-eigenvalue estimation of fourth-order partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[2]

S. ChiritaA. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[3]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[4]

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.

[5]

W. DingL. Qi and Y. Wei, $M$-tensors and nonsingular $M$-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.

[6]

W. DingJ. LiuL. Qi and H. Yan, Elasticity $M$-tensors and the strong ellipticity condition, Appl. Math. Comput., 373 (2020), 124982.  doi: 10.1016/j.amc.2019.124982.

[7]

A. DohertyP. Parillo and M. Spedalieri, Distinguishing separable and entangled states, Phys. Rev. Lett., 88 (2002), 187904.  doi: 10.1103/PhysRevLett.88.187904.

[8]

D. HanH. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[9]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[10]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-) deffiniteness of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.

[11]

S. Li and Y. Li, Bounds for the $M$-spectral radius of a fourth-order paritally symmetric tensor, J. Inequal. Appl., 218 (2018), 7pp. doi: 10.1186/s13660-018-1610-5.

[12]

L. QiH. Dai and D. Han, Conditions for strong ellipticity and $M$-eigenvalues, Front. Math. China., 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[13]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017.

[14]

C. Sang, A new Brauer-type $Z$-eigenvalue inclusion set for tensors, Numer. Algorithms, 80 (2019), 781-794.  doi: 10.1007/s11075-018-0506-2.

[15]

J. Walton and J. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Non-Linear Mech., 38 (2003), 411-455.  doi: 10.1016/S0020-7462(01)00066-X.

[16]

G. WangG. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[17]

G. WangY. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multilinear Algebra, 68 (2020), 1817-1834.  doi: 10.1080/03081087.2018.1561823.

[18]

G. WangL. Sun and L. Liu, $M$-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, 2020 (2020), 2474278. 

[19]

G. Wang and Y. Zhang, $Z$-eigenvalue exclusion theorems for tensors, J. Ind. Manag. Optim., 16 (2020), 1987-1998.  doi: 10.3934/jimo.2019039.

[20]

G. WangL. Sun and X. Wang, Sharp bounds of the minimum $M$-eigenvalue of elasticity $Z$-tensors and identifying strong ellipticity, J. Appl. Anal. Comput., 11 (2021), 2114-2130. 

[21]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest $M$-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.

[22]

G. ZhouL. Qi and S. Wu, Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor, Front. Math. China, 8 (2013), 155-168.  doi: 10.1007/s11464-012-0268-4.

[23]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134. doi: 10.1002/nla.2134.

[24]

L. Zubov and A. Rudev, On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, ZAMM Z. Angew. Math. Mech., 96 (2016), 1096-1102.  doi: 10.1002/zamm.201500167.

show all references

References:
[1]

H. CheH. Chen and Y. Wang, On the $M$-eigenvalue estimation of fourth-order partially symmetric tensors, J. Ind. Manag. Optim., 16 (2020), 309-324.  doi: 10.3934/jimo.2018153.

[2]

S. ChiritaA. Danescu and M. Ciarletta, On the strong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.

[3]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.

[4]

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.

[5]

W. DingL. Qi and Y. Wei, $M$-tensors and nonsingular $M$-tensors, Linear Algebra Appl., 439 (2013), 3264-3278.  doi: 10.1016/j.laa.2013.08.038.

[6]

W. DingJ. LiuL. Qi and H. Yan, Elasticity $M$-tensors and the strong ellipticity condition, Appl. Math. Comput., 373 (2020), 124982.  doi: 10.1016/j.amc.2019.124982.

[7]

A. DohertyP. Parillo and M. Spedalieri, Distinguishing separable and entangled states, Phys. Rev. Lett., 88 (2002), 187904.  doi: 10.1103/PhysRevLett.88.187904.

[8]

D. HanH. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.

[9]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numer. Linear Algebra Appl., 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[10]

C. Li and Y. Li, An eigenvalue localization set for tensors with applications to determine the positive (semi-) deffiniteness of tensors, Linear Multilinear Algebra, 64 (2016), 587-601.  doi: 10.1080/03081087.2015.1049582.

[11]

S. Li and Y. Li, Bounds for the $M$-spectral radius of a fourth-order paritally symmetric tensor, J. Inequal. Appl., 218 (2018), 7pp. doi: 10.1186/s13660-018-1610-5.

[12]

L. QiH. Dai and D. Han, Conditions for strong ellipticity and $M$-eigenvalues, Front. Math. China., 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.

[13]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017.

[14]

C. Sang, A new Brauer-type $Z$-eigenvalue inclusion set for tensors, Numer. Algorithms, 80 (2019), 781-794.  doi: 10.1007/s11075-018-0506-2.

[15]

J. Walton and J. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Non-Linear Mech., 38 (2003), 411-455.  doi: 10.1016/S0020-7462(01)00066-X.

[16]

G. WangG. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.

[17]

G. WangY. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear Multilinear Algebra, 68 (2020), 1817-1834.  doi: 10.1080/03081087.2018.1561823.

[18]

G. WangL. Sun and L. Liu, $M$-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors, Complexity, 2020 (2020), 2474278. 

[19]

G. Wang and Y. Zhang, $Z$-eigenvalue exclusion theorems for tensors, J. Ind. Manag. Optim., 16 (2020), 1987-1998.  doi: 10.3934/jimo.2019039.

[20]

G. WangL. Sun and X. Wang, Sharp bounds of the minimum $M$-eigenvalue of elasticity $Z$-tensors and identifying strong ellipticity, J. Appl. Anal. Comput., 11 (2021), 2114-2130. 

[21]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest $M$-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.

[22]

G. ZhouL. Qi and S. Wu, Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor, Front. Math. China, 8 (2013), 155-168.  doi: 10.1007/s11464-012-0268-4.

[23]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134. doi: 10.1002/nla.2134.

[24]

L. Zubov and A. Rudev, On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials, ZAMM Z. Angew. Math. Mech., 96 (2016), 1096-1102.  doi: 10.1002/zamm.201500167.

Table1 
References upper bounds
Theorem 1 of [11] $ \rho_M(\mathcal{A})\leq 12.6843 $
Theorem 2 of [11] $ \rho_M(\mathcal{A})\leq 10.2397 $
Theorem 3.1 of [1] $ \rho_M(\mathcal{A})\leq 15.7124 $
Theorem 3.3 of [1] $ \rho_M(\mathcal{A})\leq 15.7124 $
Theorem 3.5 of [1] $ \rho_M(\mathcal{A})\leq 15.4805 $
Theorem 3.2 $ \rho_M(\mathcal{A})\leq 8.1096 $
Theorem 3.3 $ \rho_M(\mathcal{A})\leq 8.1096 $
References upper bounds
Theorem 1 of [11] $ \rho_M(\mathcal{A})\leq 12.6843 $
Theorem 2 of [11] $ \rho_M(\mathcal{A})\leq 10.2397 $
Theorem 3.1 of [1] $ \rho_M(\mathcal{A})\leq 15.7124 $
Theorem 3.3 of [1] $ \rho_M(\mathcal{A})\leq 15.7124 $
Theorem 3.5 of [1] $ \rho_M(\mathcal{A})\leq 15.4805 $
Theorem 3.2 $ \rho_M(\mathcal{A})\leq 8.1096 $
Theorem 3.3 $ \rho_M(\mathcal{A})\leq 8.1096 $
Table2 
References upper bounds
Theorem 1 of [11] $ \rho_M(\mathcal{A})\leq 5.3333 $
Theorem 2 of [11] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.1 of [1] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.3 of [1] $ \rho_M(\mathcal{A})\leq 4.5776 $
Theorem 3.5 of [1] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.2 $ \rho_M(\mathcal{A})\leq 4.0000 $
Theorem 3.3 $ \rho_M(\mathcal{A})\leq 3.7686 $
References upper bounds
Theorem 1 of [11] $ \rho_M(\mathcal{A})\leq 5.3333 $
Theorem 2 of [11] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.1 of [1] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.3 of [1] $ \rho_M(\mathcal{A})\leq 4.5776 $
Theorem 3.5 of [1] $ \rho_M(\mathcal{A})\leq 4.7889 $
Theorem 3.2 $ \rho_M(\mathcal{A})\leq 4.0000 $
Theorem 3.3 $ \rho_M(\mathcal{A})\leq 3.7686 $
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