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doi: 10.3934/mfc.2021018
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Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

Yuyan Yao and  Gang Wang ,

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

*Corresponding author: Gang Wang

Received  June 2021 Revised  August 2021 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.

Keywords: Partially symmetric nonnegative tensors, maximum $ M $-eigenvalue, upper bounds.
Mathematics Subject Classification: Primary: 15A18, 15A42; Secondary: 15A69.
Citation: Yuyan Yao, Gang Wang. Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021018
References:
[1]

H. Che, H. 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.  Google Scholar

[2]

S. Chirita, A. 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.  Google Scholar

[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.  Google Scholar

[4]

G. Dahl, J. M. Leinaas, J. 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.  Google Scholar

[5]

W. Ding, L. 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.  Google Scholar

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W. Ding, J. Liu, L. 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.  Google Scholar

[7]

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

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D. Han, H. 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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Qi, H. 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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Wang, G. 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.  Google Scholar

[17]

G. Wang, Y. 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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

G. Wang, L. 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.   Google Scholar

[21]

Y. Wang, L. 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.  Google Scholar

[22]

G. Zhou, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. Che, H. 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.  Google Scholar

[2]

S. Chirita, A. 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.  Google Scholar

[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.  Google Scholar

[4]

G. Dahl, J. M. Leinaas, J. 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.  Google Scholar

[5]

W. Ding, L. 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.  Google Scholar

[6]

W. Ding, J. Liu, L. 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.  Google Scholar

[7]

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

[8]

D. Han, H. 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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Qi, H. 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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Wang, G. 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.  Google Scholar

[17]

G. Wang, Y. 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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

G. Wang, L. 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.   Google Scholar

[21]

Y. Wang, L. 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.  Google Scholar

[22]

G. Zhou, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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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