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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  March 2021 Early access  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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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

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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

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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