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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. |
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