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Note on $ Z $-eigenvalue inclusion theorems for tensors
School of Mathematics and Statistics, Yunnan University, Kunming 650091, China |
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.
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. |
| [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. |
| [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. |
| [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. |
| [5] |
J. He,
Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [5] |
J. He,
Bounds for the largest eigenvalue of nonnegative tensors, J. Comput. Anal. Appl., 20 (2016), 1290-1301.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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