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March  2021, 17(2): 687-693. doi: 10.3934/jimo.2019129

Note on $ Z $-eigenvalue inclusion theorems for tensors

Chaoqian Li , Yajun Liu and  Yaotang Li

School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

Received  January 2019 Revised  April 2019 Published  October 2019

Wang et al. gave four $ Z $-eigenvalue inclusion intervals for tensors in [Discrete and Continuous Dynamical Systems Series B, 1 (2017), 187-198]. However, these intervals always include zero, and hence could not be used to identify the positive definiteness of a homogeneous polynomial form. In this note, we present a new $ Z $-eigenvalue inclusion interval with parameters for even-order tensors, which not only overcomes the above shortcomings under certain conditions, but also provides a checkable sufficient condition for the positive definiteness of homogeneous polynomial forms, as well as the asymptotically stability of time-invariant polynomial systems.

Keywords: Z-eigenvalue, inclusion interval, asymptotically stable, polynomial systems.
Mathematics Subject Classification: Primary: 15A18, 15A69, 65F15, 65F10.
Citation: Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
References:
[1]

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

[2]

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

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J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

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J. He and T. 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

[7]

J. He, Y. Liu and H. Ke, et al, Bounds for the Z-spectral radius of nonnegative tensors, SpringerPlus, 5 (2016). doi: 10.1186/s40064-016-3338-3.  Google Scholar

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J. He, Y. Liu, J. Tian and Z. Zhang, New sufficient condition for the positive definiteness of fourth order tensors, Mathematics, 303 (2018), 1-10.  doi: 10.3390/math6120303.  Google Scholar

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E. Kofidis and P. 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

[10]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[11]

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

[12]

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

[13]

G. Li, L. Qi and G. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numer. Linear Algebra Appl., 20 (2013), 1001-1029.  doi: 10.1002/nla.1877.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[19]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[20]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem., Math. Program., 118 (2009), 301-316.  doi: 10.1007/s10107-007-0193-6.  Google Scholar

[21]

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

[22]

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

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

P. V. D. Driessche, Reproduction numbers of infectious disease models., Infectious Disease Model., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002.  Google Scholar

[4]

O. Duchenne, F. Bach and I. S. Kweon, et al, A tensor-based algorithm for high-order graph matching, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (2011), 2383-2395. doi: 10.1109/CVPR.2009.5206619.  Google Scholar

[5]

J. He, Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.   Google Scholar

[6]

J. He and T. 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

[7]

J. He, Y. Liu and H. Ke, et al, Bounds for the Z-spectral radius of nonnegative tensors, SpringerPlus, 5 (2016). doi: 10.1186/s40064-016-3338-3.  Google Scholar

[8]

J. He, Y. Liu, J. Tian and Z. Zhang, New sufficient condition for the positive definiteness of fourth order tensors, Mathematics, 303 (2018), 1-10.  doi: 10.3390/math6120303.  Google Scholar

[9]

E. Kofidis and P. 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

[10]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 32 (2011), 1095-1124.  doi: 10.1137/100801482.  Google Scholar

[11]

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

[12]

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

[13]

G. Li, L. Qi and G. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numer. Linear Algebra Appl., 20 (2013), 1001-1029.  doi: 10.1002/nla.1877.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

L. Qi, Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines, J. Symbolic Comput., 41 (2006), 1309-1327.  doi: 10.1016/j.jsc.2006.02.011.  Google Scholar

[19]

L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974751.ch1.  Google Scholar

[20]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem., Math. Program., 118 (2009), 301-316.  doi: 10.1007/s10107-007-0193-6.  Google Scholar

[21]

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

[22]

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

[23]

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

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