Figure 1.
Comparisons of Z-eigenvalue inclusion sets
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May
2022, 18(3): 2095-2108.
doi: 10.3934/jimo.2021058
New Z-eigenvalue localization sets for tensors with applications
School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006, China |
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.
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 and Management Optimization,
2022, 18
(3)
: 2095-2108.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
Comparisons of Geršhgorin-type Z-eigenvalue inclusion sets
Table 1.
Comparisons with the existed upper bounds
| Example 2 | Example 3 | |
| 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 |
| Example 2 | Example 3 | |
| 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 |
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