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Perron vector analysis for irreducible nonnegative tensors and its applications
1. | School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China |
2. | Department of Mathematics, University of Macau, Macau, China |
In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.
References:
[1] |
K. Chang, K. Pearson and T. Zhang,
Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[2] |
K. Chang, K. Pearson and T. Zhang,
Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 33 (2011), 806-819.
doi: 10.1137/100807120. |
[3] |
K. Chang and T. Zhang,
On the uniqueness and non-uniqueness of the positive $Z$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540.
doi: 10.1016/j.jmaa.2013.04.019. |
[4] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
[5] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
[6] |
R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991.
![]() |
[7] |
S. Hu and L. Qi,
Algebraic connectivity of an even uniform hypergraph, J. Comb. Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[8] |
W. Li, L. B. Cui and M. Ng,
The perturbation bound for the Perron vector of a transition probability tensor, Numer. Linear Algebra Appl., 20 (2013), 985-1000.
doi: 10.1002/nla.1886. |
[9] |
W. Li and M. Ng,
On the limiting probability distribution of a transition probability tensor, Linear Multilin. Algebra, 62 (2014), 362-385.
doi: 10.1080/03081087.2013.777436. |
[10] |
W. Li and M. K. Ng, The perturbation bound for the spectral radius of a nonnegative tensor, Adv. Numer. Anal., 2014 (2014), 10pp.
doi: 10.1155/2014/109525. |
[11] |
W. Li and M. K. Ng,
Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335.
doi: 10.1007/s00211-014-0666-5. |
[12] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 05, vol. 1, IEEE Computer Society Press, Piscataway, NJ, 2005, 129-132. Google Scholar |
[13] |
Q. Liu, C. Li and C. Zhang,
Some inequalities on the Perron eigenvalue and eigenvectors for positive tensors, J. of Math. Inequal., 10 (2016), 405-414.
doi: 10.7153/jmi-10-31. |
[14] |
H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. |
[15] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[16] |
K. Pearson,
Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426.
|
[17] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. of Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[18] |
L. Qi,
Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.
doi: 10.1016/j.laa.2013.03.015. |
[19] |
L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Pennsylvania, 2017.
doi: 10.1137/1.9781611974751.ch1. |
[20] |
Z. Wang and W. Wu,
Bounds for the greatest eigenvalue of positive tensors, J. of Indust. and Mgmt. Optim., 10 (2014), 1031-1039.
doi: 10.3934/jimo.2014.10.1031. |
[21] |
Y. N. Yang and Q. Z. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[22] |
Q. Z. Yang and Y. N. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250.
doi: 10.1137/100813671. |
show all references
References:
[1] |
K. Chang, K. Pearson and T. Zhang,
Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.
doi: 10.4310/CMS.2008.v6.n2.a12. |
[2] |
K. Chang, K. Pearson and T. Zhang,
Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 33 (2011), 806-819.
doi: 10.1137/100807120. |
[3] |
K. Chang and T. Zhang,
On the uniqueness and non-uniqueness of the positive $Z$-eigenvector for transition probability tensors, J. Math. Anal. Appl., 408 (2013), 525-540.
doi: 10.1016/j.jmaa.2013.04.019. |
[4] |
L. De Lathauwer, B. De Moor and J. Vandewalle,
A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), 1253-1278.
doi: 10.1137/S0895479896305696. |
[5] |
S. Friedland, S. Gaubert and L. Han,
Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749.
doi: 10.1016/j.laa.2011.02.042. |
[6] |
R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991.
![]() |
[7] |
S. Hu and L. Qi,
Algebraic connectivity of an even uniform hypergraph, J. Comb. Optim., 24 (2012), 564-579.
doi: 10.1007/s10878-011-9407-1. |
[8] |
W. Li, L. B. Cui and M. Ng,
The perturbation bound for the Perron vector of a transition probability tensor, Numer. Linear Algebra Appl., 20 (2013), 985-1000.
doi: 10.1002/nla.1886. |
[9] |
W. Li and M. Ng,
On the limiting probability distribution of a transition probability tensor, Linear Multilin. Algebra, 62 (2014), 362-385.
doi: 10.1080/03081087.2013.777436. |
[10] |
W. Li and M. K. Ng, The perturbation bound for the spectral radius of a nonnegative tensor, Adv. Numer. Anal., 2014 (2014), 10pp.
doi: 10.1155/2014/109525. |
[11] |
W. Li and M. K. Ng,
Some bounds for the spectral radius of nonnegative tensors, Numer. Math., 130 (2015), 315-335.
doi: 10.1007/s00211-014-0666-5. |
[12] |
L. H. Lim, Singular values and eigenvalues of tensors: A variational approach, in Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 05, vol. 1, IEEE Computer Society Press, Piscataway, NJ, 2005, 129-132. Google Scholar |
[13] |
Q. Liu, C. Li and C. Zhang,
Some inequalities on the Perron eigenvalue and eigenvectors for positive tensors, J. of Math. Inequal., 10 (2016), 405-414.
doi: 10.7153/jmi-10-31. |
[14] |
H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. |
[15] |
M. Ng, L. Qi and G. Zhou,
Finding the largest eigenvalue of a non-negative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.
doi: 10.1137/09074838X. |
[16] |
K. Pearson,
Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426.
|
[17] |
L. Qi,
Eigenvalues of a real supersymmetric tensor, J. of Symbolic Comput., 40 (2005), 1302-1324.
doi: 10.1016/j.jsc.2005.05.007. |
[18] |
L. Qi,
Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238.
doi: 10.1016/j.laa.2013.03.015. |
[19] |
L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, Society for Industrial and Applied Mathematics, Pennsylvania, 2017.
doi: 10.1137/1.9781611974751.ch1. |
[20] |
Z. Wang and W. Wu,
Bounds for the greatest eigenvalue of positive tensors, J. of Indust. and Mgmt. Optim., 10 (2014), 1031-1039.
doi: 10.3934/jimo.2014.10.1031. |
[21] |
Y. N. Yang and Q. Z. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530.
doi: 10.1137/090778766. |
[22] |
Q. Z. Yang and Y. N. Yang,
Further results for Perron-Frobenius theorem for nonnegative tensors Ⅱ, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250.
doi: 10.1137/100813671. |



0.5575 | 0.5307 | 0.4855 | 0.5244 |
0.5575 | 0.5307 | 0.4855 | 0.5244 |
Example 2 | Example 3 | Example 4 | |
Actual value of |
0.9873 | 0.6402 | 0.6794 |
0.6300 | 0.3969 | 0.5000 | |
0.7857 | 0.2083 | 0.3077 | |
0.5848 | 0.5000 | 0.4642 | |
|
0.2808 | 0.3445 | |
(t = -5.5602) | (t = -5.7276) | (t = -5.0250) | |
0.9258 | 0.5000 | ||
(also in (13)) | (t = -5.1168) | (t = -2.2315) | (t = -2.2956) |
|
0.6300 | 0.5503 | |
(t = -3.0000) | (t = -3.5887) | (t = -2.5208) | |
Example 2 | Example 3 | Example 4 | |
Actual value of |
0.9873 | 0.6402 | 0.6794 |
0.6300 | 0.3969 | 0.5000 | |
0.7857 | 0.2083 | 0.3077 | |
0.5848 | 0.5000 | 0.4642 | |
|
0.2808 | 0.3445 | |
(t = -5.5602) | (t = -5.7276) | (t = -5.0250) | |
0.9258 | 0.5000 | ||
(also in (13)) | (t = -5.1168) | (t = -2.2315) | (t = -2.2956) |
|
0.6300 | 0.5503 | |
(t = -3.0000) | (t = -3.5887) | (t = -2.5208) | |
Dimension | ||||
Lower bound | 42.86% | 64.02% | 75.64% | 81.37% |
Upper bound | 91.76% | 94.50% | 95.77% | 96.83% |
Dimension | ||||
Lower bound | 42.86% | 64.02% | 75.64% | 81.37% |
Upper bound | 91.76% | 94.50% | 95.77% | 96.83% |
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