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