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doi: 10.3934/jimo.2019097

## 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

* Corresponding author: Wei-Hui Liu

Received  January 2019 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by NSFC grant 11671158, U1811464 and 11771159. The second author is supported by NSFC grant 11571124 and UM grant MYRG2016-00077-FST. The third author is supported by UM grant MYRG2017-00098-FST

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.

Citation: Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019097
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. Google Scholar [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. Google Scholar [16] K. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426. Google Scholar [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. Google Scholar [18] L. Qi, Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238. doi: 10.1016/j.laa.2013.03.015. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, UK, 1991. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988. Google Scholar [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. Google Scholar [16] K. Pearson, Essentially positive tensors, Int. J. Algebra, 4 (2010), 421-426. Google Scholar [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. Google Scholar [18] L. Qi, Symmetric nonnegative tensor and copositive tensors, Linear Algebra Appl., 439 (2013), 228-238. doi: 10.1016/j.laa.2013.03.015. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
Comparison between two Ky Fan type Theorems
The results of randomly constructed tensors
The results of perturbation bounds (left) n = 5 and (right) n = 10
Comparisons with the upper bounds for the ratio
 $\omega_1$ in (3) $\omega_2$ in (4) $\omega_3$ in (5) $\omega_4$ in (11) 0.5575 0.5307 0.4855 0.5244
 $\omega_1$ in (3) $\omega_2$ in (4) $\omega_3$ in (5) $\omega_4$ in (11) 0.5575 0.5307 0.4855 0.5244
Comparisons with the lower bounds for ratio
 Example 2 Example 3 Example 4 Actual value of $\frac{x_{\min}}{x_{\max}}$ 0.9873 0.6402 0.6794 $\kappa_0$ in (2) 0.6300 0.3969 0.5000 $\kappa_1$ in (3) 0.7857 0.2083 0.3077 $\kappa_2$ in (4) 0.5848 0.5000 0.4642 $\kappa_3^{(1)}$ in (13) ${\bf{0.9662}}$ 0.2808 0.3445 (t = -5.5602) (t = -5.7276) (t = -5.0250) $\kappa_3^{(2)}$ in (16) 0.9258 0.5000 ${\bf{0.5539}}$ (also in (13)) (t = -5.1168) (t = -2.2315) (t = -2.2956) $\kappa_3^{(3)}$ in (13) 0.6300 ${\bf{0.5724}}$ 0.5503 (t = -3.0000) (t = -3.5887) (t = -2.5208) $\kappa_3$ in (13) ${\bf{0.9662}}$ ${\bf{0.5724}}$ ${\bf{0.5539}}$
 Example 2 Example 3 Example 4 Actual value of $\frac{x_{\min}}{x_{\max}}$ 0.9873 0.6402 0.6794 $\kappa_0$ in (2) 0.6300 0.3969 0.5000 $\kappa_1$ in (3) 0.7857 0.2083 0.3077 $\kappa_2$ in (4) 0.5848 0.5000 0.4642 $\kappa_3^{(1)}$ in (13) ${\bf{0.9662}}$ 0.2808 0.3445 (t = -5.5602) (t = -5.7276) (t = -5.0250) $\kappa_3^{(2)}$ in (16) 0.9258 0.5000 ${\bf{0.5539}}$ (also in (13)) (t = -5.1168) (t = -2.2315) (t = -2.2956) $\kappa_3^{(3)}$ in (13) 0.6300 ${\bf{0.5724}}$ 0.5503 (t = -3.0000) (t = -3.5887) (t = -2.5208) $\kappa_3$ in (13) ${\bf{0.9662}}$ ${\bf{0.5724}}$ ${\bf{0.5539}}$
Comparisons between (20) and (27)
 Dimension $n = 5$ $n = 10$ $n = 15$ $n = 20$ Lower bound 42.86% 64.02% 75.64% 81.37% Upper bound 91.76% 94.50% 95.77% 96.83%
 Dimension $n = 5$ $n = 10$ $n = 15$ $n = 20$ Lower bound 42.86% 64.02% 75.64% 81.37% Upper bound 91.76% 94.50% 95.77% 96.83%
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