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doi: 10.3934/jimo.2021058
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New Z-eigenvalue localization sets for tensors with applications

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.

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. BuY. 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. ChangK. 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. DengH. 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. HeY. 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. LiA. 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. LiY. 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. LiY. 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. LiW. 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. LiD. 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. 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.  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. 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.  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. BuY. 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. ChangK. 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. DengH. 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. HeY. 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. LiA. 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. LiY. 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. LiY. 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. LiW. 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. LiD. 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. 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.  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. 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.  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
Figure 2.  Comparisons of Geršhgorin-type Z-eigenvalue inclusion sets
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 $
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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