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

Z-Eigenvalue Inclusion Theorems for Tensors

Gang Wang 1, , Guanglu Zhou 2,, and  Louis Caccetta 2,

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.

Keywords: Z-eigenvalue inclusion sets, H-eigenvalue inclusion sets, weakly symmetric nonnegative tensors, largest Z-eigenvalue, spectral radius.
Mathematics Subject Classification: Primary:15A18, 15A42;Secondary:15A6.
Citation: Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete & 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.  Google Scholar

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

[3]

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.   Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[9]

Y. Liu, G. 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.   Google Scholar

[10]

G. Li, L. 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.   Google Scholar

[11]

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

[12]

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

[13]

Q. Ni, L. 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.   Google Scholar

[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.  Google Scholar

[15]

L. Qi, G. 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.  Google Scholar

[16]

L. Qi, F. 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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

G. Zhou, L. 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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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.   Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

Y. Liu, G. 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.   Google Scholar

[10]

G. Li, L. 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.   Google Scholar

[11]

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

[12]

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

[13]

Q. Ni, L. 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.   Google Scholar

[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.  Google Scholar

[15]

L. Qi, G. 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.  Google Scholar

[16]

L. Qi, F. 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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

G. Zhou, L. 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.  Google Scholar

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}\}$$\mathcal{L}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 5\}$
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$\mathcal{L}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$$\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\}$.
Table Options

Download as excel

$\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\}$.
Table Options

Download as excel

$\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\}$.
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