-
Previous Article
Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances
- NACO Home
- This Issue
-
Next Article
An optimal control problem by parabolic equation with boundary smooth control and an integral constraint
New bounds for eigenvalues of strictly diagonally dominant tensors
School of Mathematical Sciences, Tianjin University, Tianjin 300350, China |
In this paper, we prove that the minimum eigenvalue of a strictly diagonally dominant Z-tensor with positive diagonal entries lies between the smallest and the largest row sums. The novelty comes from the upper bound. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries. Furthermore, other new bounds for the minimum eigenvalue of nonsingular M-tensors are obtained.
References:
| [1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8. |
| [2] |
K. C. Chang, K. Pearson and T. Zhang,
Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra and Its Applications, 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
| [3] |
K. C. Chang, K. Pearson and T. Zhang,
Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.
|
| [4] |
W Ding, L. Qi and Y. Wei,
M-Tensors and nonsingular M-Tensors, Linear Algebra and Its Applications, 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
| [5] |
J. He and Z. Huang,
Upper bound for the largest Z-eigenvalue of positive tensors, Linear Algebra and Its Applications, 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
| [6] |
J. He and Z. Huang, Inequalities for M-tensors Journal of Inequalities and Applications, (2014), 114, 9 pages.
doi: 10.1186/1029-242X-2014-114. |
| [7] |
L. De Lathauwer, B. D. Moor and J. Vandewalle,
On the best rank-1 and rank-($R_{1},R_{2},...,R_{N}$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
| [8] |
C. Li, Y. Li and K. Xu,
New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
| [9] |
L. H. Lim,
Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, (2005), 129-132.
|
| [10] |
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. |
| [11] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [12] |
L. Qi, W. Sun and Y. Wang,
Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.
doi: 10.1007/s11464-007-0031-4. |
| [13] |
L. Qi, Y. Wang and E. X. Wu,
D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.
doi: 10.1016/j.cam.2007.10.012. |
| [14] |
F. Wang, The tensor eigenvalue methods for the positive definiteness identification problem, Available at http://ira.lib.polyu.edu.hk/handle/10397/2642, Hong Kong Polytechnic University, 2006. |
| [15] |
Q. Yang and Y. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1236-1250.
doi: 10.1137/100813671. |
| [16] |
Y. Yang and Q. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2011), 2517-2530.
doi: 10.1137/090778766. |
| [17] |
L. Zhang, L. Qi and G. Zhou, M-tensors and the positive definiteness of a multivariate form, Mathematics, 2012. |
| [18] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 437-452.
doi: 10.1137/130915339. |
show all references
References:
| [1] |
L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, In: Medical Image Computing and Computer-Assisted Intervention-MICCAI 2008, Springer, Berlin/Heidelberg, (2008), 1-8. |
| [2] |
K. C. Chang, K. Pearson and T. Zhang,
Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra and Its Applications, 438 (2013), 4166-4182.
doi: 10.1016/j.laa.2013.02.013. |
| [3] |
K. C. Chang, K. Pearson and T. Zhang,
Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.
|
| [4] |
W Ding, L. Qi and Y. Wei,
M-Tensors and nonsingular M-Tensors, Linear Algebra and Its Applications, 439 (2013), 3264-3278.
doi: 10.1016/j.laa.2013.08.038. |
| [5] |
J. He and Z. Huang,
Upper bound for the largest Z-eigenvalue of positive tensors, Linear Algebra and Its Applications, 38 (2014), 110-114.
doi: 10.1016/j.aml.2014.07.012. |
| [6] |
J. He and Z. Huang, Inequalities for M-tensors Journal of Inequalities and Applications, (2014), 114, 9 pages.
doi: 10.1186/1029-242X-2014-114. |
| [7] |
L. De Lathauwer, B. D. Moor and J. Vandewalle,
On the best rank-1 and rank-($R_{1},R_{2},...,R_{N}$) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.
doi: 10.1137/S0895479898346995. |
| [8] |
C. Li, Y. Li and K. Xu,
New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.
doi: 10.1002/nla.1858. |
| [9] |
L. H. Lim,
Singular values and eigenvalues of tensors: A variational approach, in CAMSAP'05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-tensor Adaptive Processing, (2005), 129-132.
|
| [10] |
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. |
| [11] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
| [12] |
L. Qi, W. Sun and Y. Wang,
Numerical multilinear algebra and its applications, Front. Math. China, 2 (2007), 501-526.
doi: 10.1007/s11464-007-0031-4. |
| [13] |
L. Qi, Y. Wang and E. X. Wu,
D-eigenvalues of diffusion kurtosis tensor, Journal of Computational and Applied Mathematics, 221 (2008), 150-157.
doi: 10.1016/j.cam.2007.10.012. |
| [14] |
F. Wang, The tensor eigenvalue methods for the positive definiteness identification problem, Available at http://ira.lib.polyu.edu.hk/handle/10397/2642, Hong Kong Polytechnic University, 2006. |
| [15] |
Q. Yang and Y. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1236-1250.
doi: 10.1137/100813671. |
| [16] |
Y. Yang and Q. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2011), 2517-2530.
doi: 10.1137/090778766. |
| [17] |
L. Zhang, L. Qi and G. Zhou, M-tensors and the positive definiteness of a multivariate form, Mathematics, 2012. |
| [18] |
L. Zhang, L. Qi and G. Zhou,
M-tensors and some applications, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 437-452.
doi: 10.1137/130915339. |
| [1] |
Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial and Management Optimization, 2020, 16 (6) : 3035-3045. doi: 10.3934/jimo.2019092 |
| [2] |
Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2551-2562. doi: 10.3934/jimo.2019069 |
| [3] |
Chong Wang, Gang Wang, Lixia Liu. Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021205 |
| [4] |
Ruixue Zhao, Jinyan Fan. Quadratic tensor eigenvalue complementarity problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022073 |
| [5] |
Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042 |
| [6] |
Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153 |
| [7] |
Fan Wu. Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor. Evolution Equations and Control Theory, 2021, 10 (3) : 511-518. doi: 10.3934/eect.2020078 |
| [8] |
Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial and Management Optimization, 2022, 18 (1) : 157-172. doi: 10.3934/jimo.2020147 |
| [9] |
Haitao Che, Haibin Chen, Guanglu Zhou. New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3685-3694. doi: 10.3934/jimo.2020139 |
| [10] |
Yuyan Yao, Gang Wang. Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors. Mathematical Foundations of Computing, 2022, 5 (1) : 33-44. doi: 10.3934/mfc.2021018 |
| [11] |
Kaiping Liu, Haitao Che, Haibin Chen, Meixia Li. Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022077 |
| [12] |
Ian Blake, V. Kumar Murty, Hamid Usefi. A note on diagonal and Hermitian hypersurfaces. Advances in Mathematics of Communications, 2016, 10 (4) : 753-764. doi: 10.3934/amc.2016039 |
| [13] |
Grzegorz Siudem, Grzegorz Świątek. Diagonal stationary points of the bethe functional. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2717-2743. doi: 10.3934/dcds.2017117 |
| [14] |
Vasso Anagnostopoulou. Stochastic dominance for shift-invariant measures. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 667-682. doi: 10.3934/dcds.2019027 |
| [15] |
Debasisha Mishra. Matrix group monotonicity using a dominance notion. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 267-274. doi: 10.3934/naco.2015.5.267 |
| [16] |
Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029 |
| [17] |
Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020130 |
| [18] |
Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. M/M/3/3 and M/M/4/4 retrial queues. Journal of Industrial and Management Optimization, 2009, 5 (3) : 431-451. doi: 10.3934/jimo.2009.5.431 |
| [19] |
Jan Boman, Vladimir Sharafutdinov. Stability estimates in tensor tomography. Inverse Problems and Imaging, 2018, 12 (5) : 1245-1262. doi: 10.3934/ipi.2018052 |
| [20] |
Shenglong Hu. A note on the solvability of a tensor equation. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021146 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]