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July  2020, 16(4): 1987-1998. doi: 10.3934/jimo.2019039

$ Z $-eigenvalue exclusion theorems for tensors

Gang Wang , and  Yuan Zhang

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

* Corresponding author: Gang Wang

Received  July 2018 Revised  November 2018 Published  July 2020 Early access  May 2019

Fund Project: This work was supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10)

To locate all $ Z$-eigenvalues of a tensor more precisely, we establish three $Z$-eigenvalue exclusion sets such that all $ Z$-eigenvalues do not belong to them and get three tighter $Z$-eigenvalue inclusion sets of tensor by using these $Z$-eigenvalue exclusion sets. Furthermore, we show that the new inclusion sets are tighter than the existing results via two running examples.

Keywords: Tensor eigenvalue, Z-eigenvalue inclusion sets, Z-eigenvalue exclusion sets.
Mathematics Subject Classification: Primary: 15A18, 15A42.
Citation: Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1987-1998. doi: 10.3934/jimo.2019039
References:
[1]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, (2008), 1–8. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

[2]

K. Chang, K. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[3]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[4]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with applications, 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[5]

L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 21 (2000), 1253-1278.  doi: 10.1137/S0895479896305696.  Google Scholar

[6]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[7]

L. Gao, D. Wang and G. Wang, Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Applied Mathematics and Computation, 268 (2015), 186-200.  doi: 10.1016/j.amc.2015.06.023.  Google Scholar

[8]

L. Gao, D. Wang and G. Zong., Exponential stability for generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching, Nonlinear Analysis: Hybrid Systems, 30 (2018), 199-212.  doi: 10.1016/j.nahs.2018.05.009.  Google Scholar

[9]

J. He and T. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[10]

E. Kofidis and R. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[11]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 34 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[12]

W. Li and M. Ng, On the Limiting probability distribution of a transition probability tensor, Linear Multilinear Algebra, 62 (2014), 362-385.  doi: 10.1080/03081087.2013.777436.  Google Scholar

[13]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bound for an irreducible nonnegative tensor, Linear Algebra and its Applications, 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[14]

C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, Journal of Computational and Applied Mathematics, 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.  Google Scholar

[15]

C. Li, S. Li, Q. Liu and Y. Li, Exclusion sets in eigenvalue inclusion sets for tensors, arXiv: 1706.00944. Google Scholar

[16]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach., Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129–132. Google Scholar

[17]

Q. Liu and Y. Li, Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Mathematics, 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[18]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[19]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.  Google Scholar

[20]

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

[21]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[22]

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

[23]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[24]

G. Wang, G. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[25]

G. Wang, G. Zhou and L. Caccetta, Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pacific journal of Optimization, 14 (2018), 227-244.   Google Scholar

[26]

G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, (2019). doi: 10.1080/03081087.2018.1561823.  Google Scholar

[27]

G. Wang, Y. Wang and L. Liu, Bound estimations on the eigenvalues for Fan product of $M $-tensors, Taiwanese Journal of Mathematics, (2019), 16pp. doi: 10.11650/tjm/180905.  Google Scholar

[28]

Y. Wang and G. Wang, Two S-type $ Z$-eigenvalue inclusion sets for tensors, Journal of Inequalities and Application, 2017 (2017), 1-12.  doi: 10.1186/s13660-017-1428-6.  Google Scholar

[29]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[30]

Y. Wang, G. Zhou and L. Caccetta, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[31]

Y. Wang, K. Zhang and H. Sun, Criteria for strong $ H$-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[32]

Y. Wang, G. Zhou and L. Caccetta, Nonsingular $ H$-tensor and its criteria, Journal of Industrial and Management Optimization, 12 (2016), 1173-1186.  doi: 10.3934/jimo.2016.12.1173.  Google Scholar

[33]

K. Zhang and Y. Wang, An $H $-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[34]

J. Zhao and C. Sang, A new $ Z$-eigenvalue localization set for tensors, Journal of Inequalities and Application, 2017 (2017), 1-9.  doi: 10.1186/s13660-017-1363-6.  Google Scholar

[35]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with applications, 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

show all references

References:
[1]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, in Medical Image Computing and Computer-Assisted Intervention, Springer, (2008), 1–8. doi: 10.1007/978-3-540-85988-8_1.  Google Scholar

[2]

K. Chang, K. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[3]

H. Chen, L. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Frontiers of Mathematics in China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[4]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numerical Linear Algebra with applications, 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[5]

L. De Lathauwer, B. De Moor and J. Vandewalle, A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 21 (2000), 1253-1278.  doi: 10.1137/S0895479896305696.  Google Scholar

[6]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749.  doi: 10.1016/j.laa.2011.02.042.  Google Scholar

[7]

L. Gao, D. Wang and G. Wang, Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects, Applied Mathematics and Computation, 268 (2015), 186-200.  doi: 10.1016/j.amc.2015.06.023.  Google Scholar

[8]

L. Gao, D. Wang and G. Zong., Exponential stability for generalized stochastic impulsive functional differential equations with delayed impulses and Markovian switching, Nonlinear Analysis: Hybrid Systems, 30 (2018), 199-212.  doi: 10.1016/j.nahs.2018.05.009.  Google Scholar

[9]

J. He and T. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[10]

E. Kofidis and R. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[11]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 34 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[12]

W. Li and M. Ng, On the Limiting probability distribution of a transition probability tensor, Linear Multilinear Algebra, 62 (2014), 362-385.  doi: 10.1080/03081087.2013.777436.  Google Scholar

[13]

W. Li, D. Liu and S. W. Vong, $Z$-eigenpair bound for an irreducible nonnegative tensor, Linear Algebra and its Applications, 483 (2015), 182-199.  doi: 10.1016/j.laa.2015.05.033.  Google Scholar

[14]

C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, Journal of Computational and Applied Mathematics, 255 (2014), 1-14.  doi: 10.1016/j.cam.2013.04.022.  Google Scholar

[15]

C. Li, S. Li, Q. Liu and Y. Li, Exclusion sets in eigenvalue inclusion sets for tensors, arXiv: 1706.00944. Google Scholar

[16]

L. H. Lim, Singular values and eigenvalues of tensors: A variational approach., Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129–132. Google Scholar

[17]

Q. Liu and Y. Li, Bounds for the $ Z$-eigenpair of general nonnegative tensors, Open Mathematics, 14 (2016), 181-194.  doi: 10.1515/math-2016-0017.  Google Scholar

[18]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099.  doi: 10.1137/09074838X.  Google Scholar

[19]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107.  doi: 10.1109/TAC.2008.923679.  Google Scholar

[20]

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

[21]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[22]

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

[23]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[24]

G. Wang, G. Zhou and L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[25]

G. Wang, G. Zhou and L. Caccetta, Sharp Brauer-type eigenvalue inclusion theorems for tensors, Pacific journal of Optimization, 14 (2018), 227-244.   Google Scholar

[26]

G. Wang, Y. Wang and Y. Wang, Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors, Linear and Multilinear Algebra, (2019). doi: 10.1080/03081087.2018.1561823.  Google Scholar

[27]

G. Wang, Y. Wang and L. Liu, Bound estimations on the eigenvalues for Fan product of $M $-tensors, Taiwanese Journal of Mathematics, (2019), 16pp. doi: 10.11650/tjm/180905.  Google Scholar

[28]

Y. Wang and G. Wang, Two S-type $ Z$-eigenvalue inclusion sets for tensors, Journal of Inequalities and Application, 2017 (2017), 1-12.  doi: 10.1186/s13660-017-1428-6.  Google Scholar

[29]

X. Wang, H. Chen and Y. Wang, Solution structures of tensor complementarity problem, Frontiers of Mathematics in China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[30]

Y. Wang, G. Zhou and L. Caccetta, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numerical Linear Algebra with applications, 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[31]

Y. Wang, K. Zhang and H. Sun, Criteria for strong $ H$-tensors, Frontiers of Mathematics in China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[32]

Y. Wang, G. Zhou and L. Caccetta, Nonsingular $ H$-tensor and its criteria, Journal of Industrial and Management Optimization, 12 (2016), 1173-1186.  doi: 10.3934/jimo.2016.12.1173.  Google Scholar

[33]

K. Zhang and Y. Wang, An $H $-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, Journal of Computational and Applied Mathematics, 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[34]

J. Zhao and C. Sang, A new $ Z$-eigenvalue localization set for tensors, Journal of Inequalities and Application, 2017 (2017), 1-9.  doi: 10.1186/s13660-017-1363-6.  Google Scholar

[35]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numerical Linear Algebra with applications, 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

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