New Z-eigenvalue localization sets for tensors with applications

Jun He; Guangjun Xu; Yanmin Liu

doi:10.3934/jimo.2021058

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doi: 10.3934/jimo.2021058

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New Z-eigenvalue localization sets for tensors with applications

Jun He ^, , Guangjun Xu and Yanmin Liu

School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006, China

^* Corresponding author: Jun He

Received June 2020 Revised January 2021 Early access March 2021

Fund Project: This work is supported by NSF of China (71461027, 11661084), Innovative talent team in Guizhou Province (Qian Ke He Pingtai Rencai[2016]5619), New academic talents and innovative exploration fostering project(Qian Ke He Pingtai Rencai[2017]5727-21), Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2020]094)

In this paper, let $ \mathcal{A} = (a_{i_1 i_2 \cdots i_m } )\in{\mathbb{R}}^{[m, n]} $, when $ m\geq 4 $, based on the condition $ ||x||_2 = 1 $, a new $ Z $-eigenvalue localization set for tensors is given. And we extend the Geršhgorin-type localization set for Z-eigenvalues of fourth order tensors to higher order tensors. As an application, a sharper upper bound for the Z-spectral radius of nonnegative tensors is obtained. Let $ \mathcal{H} $ be a $ k $-uniform hypergraph with $ k\geq 4 $ and $ \mathcal{A}(\mathcal{H}) $ be the adjacency tensor of $ \mathcal{H} $, a new upper bound for the Z-spectral radius $ \rho (\mathcal{H}) $ is also presented. Finally, a checkable sufficient condition for the positive definiteness of even-order tensors and asymptotically stability of time-invariant polynomial systems is also given.

Keywords: Localization set, bound, Z-eigenvalue, asymptotically stability.

Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 65F15, 65F10.

Citation: Jun He, Guangjun Xu, Yanmin Liu. New Z-eigenvalue localization sets for tensors with applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021058

References:

[1]	C. Bu, Y. Fan and J. Zhou, Laplacian and signless Laplacian $Z$-eigenvalues of uniform hypergraphs, Front. Math. China, 11 (2016), 511-520. doi: 10.1007/s11464-015-0467-x. Google Scholar
[2]	K. C. Chang, K. J. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182. doi: 10.1016/j.laa.2013.02.013. Google Scholar
[3]	K. C. Chang and T. Zhang, On the uniqueness and non-uniqueness of the positive ${Z}$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540. doi: 10.1016/j.jmaa.2013.04.019. Google Scholar
[4]	C. Deng, H. Li and C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl., 556 (2018), 55-69. doi: 10.1016/j.laa.2018.06.032. Google Scholar
[5]	J. He and T.-Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114. doi: 10.1016/j.aml.2014.07.012. Google Scholar
[6]	J. He, Y. Liu and G. Xu, $Z$-eigenvalues-based sufficient conditions for the positive deffiniteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 43 (2020), 1069-1093. doi: 10.1007/s40840-019-00727-7. Google Scholar
[7]	E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884. doi: 10.1137/S0895479801387413. Google Scholar
[8]	T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124. doi: 10.1137/100801482. Google Scholar
[9]	C. Li, A. Jiao and Y. Li, An $S$-type eigenvalue localization set for tensors, Linear Algebra Appl., 493 (2016), 469-483. doi: 10.1016/j.laa.2015.12.018. Google Scholar
[10]	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
[11]	C. Li, Y. Liu and Y. Li, Note on $Z$-eigenvalue inclusion theorems for tensors, J. Ind. Manag. Optim., 17 (2021), 687-693. doi: 10.3934/jimo.2019129. Google Scholar
[12]	W. Li, W. Liu and S.-W. Vong, Some bounds for $H$-eigenpairs and $Z$-eigenpairs of a tensor, J. Comput. Appl. Math., 342 (2018), 37-57. doi: 10.1016/j.cam.2018.03.024. Google Scholar
[13]	W. Li, D. Liu and S.-W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199. doi: 10.1016/j.laa.2015.05.033. Google Scholar
[14]	Q. Liu and Y. Li, Bounds for the $Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194. doi: 10.1515/math-2016-0017. Google Scholar
[15]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. Google Scholar
[16]	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
[17]	C. Sang and Z. Chen, $Z$-eigenvalue localization sets for even order tensors and their applications, Acta Appl. Math., 169 (2020) 323–339. doi: 10.1007/s10440-019-00300-1. Google Scholar
[18]	Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595. doi: 10.1137/130909135. Google Scholar
[19]	L. Sun, G. 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. Google Scholar
[20]	Y. Wang and G. Wang, Two S-type $S$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), 152, 12 pp. doi: 10.1186/s13660-017-1428-6. Google Scholar
[21]	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
[22]	J. Xie and A. Chang, On the $Z$-eigenvalues of the adjacency tensors for uniform hypergraphs, Linear Algebra Appl., 439 (2013), 2195-2204. doi: 10.1016/j.laa.2013.07.016. Google Scholar
[23]	J. Xie and A. Chang, On the $Z$-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph, Numer. Linear Algebra Appl., 20 (2013), 1030-1045. doi: 10.1002/nla.1910. Google Scholar
[24]	J. Zhao, A new $Z$–eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), 85, 9 pp. doi: 10.1186/s13660-017-1363-6. Google Scholar

show all references

References:

[1]	C. Bu, Y. Fan and J. Zhou, Laplacian and signless Laplacian $Z$-eigenvalues of uniform hypergraphs, Front. Math. China, 11 (2016), 511-520. doi: 10.1007/s11464-015-0467-x. Google Scholar
[2]	K. C. Chang, K. J. Pearson and T. Zhang, Some variational principles for $Z$-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182. doi: 10.1016/j.laa.2013.02.013. Google Scholar
[3]	K. C. Chang and T. Zhang, On the uniqueness and non-uniqueness of the positive ${Z}$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540. doi: 10.1016/j.jmaa.2013.04.019. Google Scholar
[4]	C. Deng, H. Li and C. Bu, Brauer-type eigenvalue inclusion sets of stochastic/irreducible tensors and positive definiteness of tensors, Linear Algebra Appl., 556 (2018), 55-69. doi: 10.1016/j.laa.2018.06.032. Google Scholar
[5]	J. He and T.-Z. Huang, Upper bound for the largest $Z$-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114. doi: 10.1016/j.aml.2014.07.012. Google Scholar
[6]	J. He, Y. Liu and G. Xu, $Z$-eigenvalues-based sufficient conditions for the positive deffiniteness of fourth-order tensors, Bull. Malays. Math. Sci. Soc., 43 (2020), 1069-1093. doi: 10.1007/s40840-019-00727-7. Google Scholar
[7]	E. Kofidis and P. A. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884. doi: 10.1137/S0895479801387413. Google Scholar
[8]	T. G. Kolda and J. R. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124. doi: 10.1137/100801482. Google Scholar
[9]	C. Li, A. Jiao and Y. Li, An $S$-type eigenvalue localization set for tensors, Linear Algebra Appl., 493 (2016), 469-483. doi: 10.1016/j.laa.2015.12.018. Google Scholar
[10]	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
[11]	C. Li, Y. Liu and Y. Li, Note on $Z$-eigenvalue inclusion theorems for tensors, J. Ind. Manag. Optim., 17 (2021), 687-693. doi: 10.3934/jimo.2019129. Google Scholar
[12]	W. Li, W. Liu and S.-W. Vong, Some bounds for $H$-eigenpairs and $Z$-eigenpairs of a tensor, J. Comput. Appl. Math., 342 (2018), 37-57. doi: 10.1016/j.cam.2018.03.024. Google Scholar
[13]	W. Li, D. Liu and S.-W. Vong, $Z$-eigenpair bounds for an irreducible nonnegative tensor, Linear Algebra Appl., 483 (2015), 182-199. doi: 10.1016/j.laa.2015.05.033. Google Scholar
[14]	Q. Liu and Y. Li, Bounds for the $Z$-eigenpair of general nonnegative tensors, Open Math., 14 (2016), 181-194. doi: 10.1515/math-2016-0017. Google Scholar
[15]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007. Google Scholar
[16]	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
[17]	C. Sang and Z. Chen, $Z$-eigenvalue localization sets for even order tensors and their applications, Acta Appl. Math., 169 (2020) 323–339. doi: 10.1007/s10440-019-00300-1. Google Scholar
[18]	Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595. doi: 10.1137/130909135. Google Scholar
[19]	L. Sun, G. 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. Google Scholar
[20]	Y. Wang and G. Wang, Two S-type $S$-eigenvalue inclusion sets for tensors, J. Inequal. Appl., 2017 (2017), 152, 12 pp. doi: 10.1186/s13660-017-1428-6. Google Scholar
[21]	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
[22]	J. Xie and A. Chang, On the $Z$-eigenvalues of the adjacency tensors for uniform hypergraphs, Linear Algebra Appl., 439 (2013), 2195-2204. doi: 10.1016/j.laa.2013.07.016. Google Scholar
[23]	J. Xie and A. Chang, On the $Z$-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph, Numer. Linear Algebra Appl., 20 (2013), 1030-1045. doi: 10.1002/nla.1910. Google Scholar
[24]	J. Zhao, A new $Z$–eigenvalue localization set for tensors, J. Inequal. Appl., 2017 (2017), 85, 9 pp. doi: 10.1186/s13660-017-1363-6. Google Scholar

Figure 1. Comparisons of Z-eigenvalue inclusion sets

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Figure 2. Comparisons of Geršhgorin-type Z-eigenvalue inclusion sets

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Table 1. Comparisons with the existed upper bounds

	Example 2	Example 3
$ \rho (\mathcal{A}) $	3.1092	7.3525
Bound (2)	5.3333	40
Bound (3)	5.0437	25
Bound (4)	5.2846	-
Bound (5)	5.1935	-
Theorem 4.2 of [12]	4.4632	-
Theorem 3.1	4	$ 20 $

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	Example 2	Example 3
$ \rho (\mathcal{A}) $	3.1092	7.3525
Bound (2)	5.3333	40
Bound (3)	5.0437	25
Bound (4)	5.2846	-
Bound (5)	5.1935	-
Theorem 4.2 of [12]	4.4632	-
Theorem 3.1	4	$ 20 $

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