January  2017, 22(1): 187-198. doi: 10.3934/dcdsb.2017009

Z-Eigenvalue Inclusion Theorems for Tensors

1. 

School of Management Science, Qufu Normal University, Rizhao, Shandong 276826, China

2. 

Department of Mathematics and Statistics, Curtin University, Perth, Australia

* Corresponding author: Guanglu Zhou

Received  December 2015 Revised  May 2016 Published  December 2016

Fund Project: The first author is supported by the natural science foundation of Shandong Province grant ZR2016AM10 and the Fundamental Research Funds for Qufu Normal University grant xkj201415, xkj201314.

In this paper, we establish $Z$-eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between $Z$-eigenvalues and $H$-eigenvalues. As an application, we obtain upper bounds for the largest $Z$-eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

Citation: Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 187-198. doi: 10.3934/dcdsb.2017009
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, Springer, 5241 (2008), 1-8.  doi: 10.1007/978-3-540-85988-8_1.

[2]

K. C. ChangK. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Communications in Mathematical Sciences, 6 (2008), 507-520. 

[3]

K. C. ChangK. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182. 

[4]

S. FriedlandS. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749. 

[5]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.

[6]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095-1124.  doi: 10.1137/100801482.

[7]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[8]

L. H. Lim, Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129-132. 

[9]

Y. LiuG. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, Journal of Computational and Applied Mathematics, 235 (2010), 286-292. 

[10]

G. LiL. Qi and G. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numerical Linear Algebra with Applications, 20 (2013), 1001-1029. 

[11]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[12]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099. 

[13]

Q. NiL. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. 

[14]

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

[15]

L. QiG. Yu and E. X. Wu, Higher order positive semi-definite diffusion tensor imaging, SIAM Journal on Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.

[16]

L. QiF. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316.  doi: 10.1007/s10107-007-0193-6.

[17]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[18]

R. S. Varga, Gergorin and His Circles Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-642-17798-9.

[19]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, Journal of Industrial and Managenment Optimization, 10 (2014), 1031-1039. 

[20]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[21]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[22]

L. Zhang and L. Qi, Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor, Numerical Linear Algebra and Its Applications, 19 (2012), 830-841. 

[23]

G. ZhouL. Qi and S. Wu, On the largest eigenvalue of a symmetric nonnegative tensor, Numerical Linear Algebra with Applications, 20 (2013), 913-928.  doi: 10.1002/nla.1885.

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, Springer, 5241 (2008), 1-8.  doi: 10.1007/978-3-540-85988-8_1.

[2]

K. C. ChangK. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Communications in Mathematical Sciences, 6 (2008), 507-520. 

[3]

K. C. ChangK. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra and its Applications, 438 (2013), 4166-4182. 

[4]

S. FriedlandS. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra and its Applications, 438 (2013), 738-749. 

[5]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.

[6]

T. Kolda and J. Mayo, Shifted power method for computing tensor eigenpairs, SIAM Journal on Matrix Analysis and Applications, 32 (2011), 1095-1124.  doi: 10.1137/100801482.

[7]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Applied Mathematics Letters, 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.

[8]

L. H. Lim, Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE International Workshop on Computational Advances, in Multi-Sensor Adaptive Processing, Puerto Vallarta, (2005), 129-132. 

[9]

Y. LiuG. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, Journal of Computational and Applied Mathematics, 235 (2010), 286-292. 

[10]

G. LiL. Qi and G. Yu, The Z-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory, Numerical Linear Algebra with Applications, 20 (2013), 1001-1029. 

[11]

C. LiY. Li and X. Kong, New eigenvalue inclusion sets for tensors, Numerical Linear Algebra with Applications, 21 (2014), 39-50.  doi: 10.1002/nla.1858.

[12]

M. NgL. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1090-1099. 

[13]

Q. NiL. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. 

[14]

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

[15]

L. QiG. Yu and E. X. Wu, Higher order positive semi-definite diffusion tensor imaging, SIAM Journal on Imaging Sciences, 3 (2010), 416-433.  doi: 10.1137/090755138.

[16]

L. QiF. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316.  doi: 10.1007/s10107-007-0193-6.

[17]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 1581-1595.  doi: 10.1137/130909135.

[18]

R. S. Varga, Gergorin and His Circles Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-642-17798-9.

[19]

Z. Wang and W. Wu, Bounds for the greatest eigenvalue of positive tensors, Journal of Industrial and Managenment Optimization, 10 (2014), 1031-1039. 

[20]

Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2517-2530.  doi: 10.1137/090778766.

[21]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM Journal on Matrix Analysis and Applications, 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.

[22]

L. Zhang and L. Qi, Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor, Numerical Linear Algebra and Its Applications, 19 (2012), 830-841. 

[23]

G. ZhouL. Qi and S. Wu, On the largest eigenvalue of a symmetric nonnegative tensor, Numerical Linear Algebra with Applications, 20 (2013), 913-928.  doi: 10.1002/nla.1885.

Table1 
$\mathcal{L}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{L}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}\}$
$\mathcal{L}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{L}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$
$\mathcal{L}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$$\mathcal{L}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 5\}$
$\mathcal{L}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{L}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}\}$
$\mathcal{L}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{L}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$
$\mathcal{L}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$$\mathcal{L}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 5\}$
Table2 
$\mathcal{M}_{1,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 4\}$$\mathcal{M}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{7+\sqrt{5}}{2}\}$
$\mathcal{M}_{2,1}(\mathcal{A})=\{\lambda\in C: 2\leq |\lambda|\leq 4\}$$\mathcal{M}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$
$\mathcal{M}_{3,1}(\mathcal{A})=\{\lambda\in C: 3+\sqrt{3}\leq |\lambda|\leq 5\}$$\mathcal{M}_{3,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 5\}$.
$\mathcal{M}_{1,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 4\}$$\mathcal{M}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{7+\sqrt{5}}{2}\}$
$\mathcal{M}_{2,1}(\mathcal{A})=\{\lambda\in C: 2\leq |\lambda|\leq 4\}$$\mathcal{M}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$
$\mathcal{M}_{3,1}(\mathcal{A})=\{\lambda\in C: 3+\sqrt{3}\leq |\lambda|\leq 5\}$$\mathcal{M}_{3,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 5\}$.
Table3 
$\mathcal{N}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{21}}{2}\}$$\mathcal{N}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2} $
$\mathcal{N}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{N}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2} $
$\mathcal{N}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$$\mathcal{N}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$.
$\mathcal{N}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{21}}{2}\}$$\mathcal{N}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2} $
$\mathcal{N}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\mathcal{N}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2} $
$\mathcal{N}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$$\mathcal{N}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$.
[1]

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

[2]

Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial and Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054

[3]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial and Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129

[4]

Caili Sang, Zhen Chen. Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2435-2468. doi: 10.3934/jimo.2021075

[5]

Yang Xu, Zheng-Hai Huang. Pareto eigenvalue inclusion intervals for tensors. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022035

[6]

Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1987-1998. doi: 10.3934/jimo.2019039

[7]

Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75

[8]

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

[9]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial and Management Optimization, 2020, 16 (4) : 2045-2063. doi: 10.3934/jimo.2019042

[10]

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

[11]

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

[12]

Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial and Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975

[13]

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

[14]

Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031

[15]

Xifu Liu, Shuheng Yin, Hanyu Li. C-eigenvalue intervals for piezoelectric-type tensors via symmetric matrices. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3349-3356. doi: 10.3934/jimo.2020122

[16]

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

[17]

Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765

[18]

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

[19]

Yannan Chen, Antal Jákli, Liqun Qi. The C-eigenvalue of third order tensors and its application in crystals. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021183

[20]

Guimin Liu, Hongbin Lv. Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021176

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (315)
  • HTML views (138)
  • Cited by (16)

Other articles
by authors

[Back to Top]