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Article Contents

# Perron vector analysis for irreducible nonnegative tensors and its applications

• * Corresponding author: Wei-Hui Liu

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.

Mathematics Subject Classification: Primary: 15A69.

 Citation:

• Figure 1.  Comparison between two Ky Fan type Theorems

Figure 2.  The results of randomly constructed tensors

Figure 3.  The results of perturbation bounds (left) n = 5 and (right) n = 10

Table 1.  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

Table 2.  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) ${\bf{0.9662}}$ ${\bf{0.5724}}$ ${\bf{0.5539}}$

Table 3.  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%
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