# American Institute of Mathematical Sciences

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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