# American Institute of Mathematical Sciences

January  2021, 17(1): 29-50. 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  January 2021 Early access  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 and Management Optimization, 2021, 17 (1) : 29-50. 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. [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. [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. [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.
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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