New Z-eigenvalue localization sets for tensors with applications

Jun He; Guangjun Xu; Yanmin Liu

doi:10.3934/jimo.2021058

Article Contents

2022, Volume 18, Issue 3: 2095-2108. Doi: 10.3934/jimo.2021058

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New Z-eigenvalue localization sets for tensors with applications

School of Mathematics, Zunyi Normal College, Zunyi, Guizhou 563006, China

^* Corresponding author: Jun He
^* Corresponding author: Jun He

Received: May 31, 2020

Revised: December 31, 2020

Published: May 2022

This work is supported by NSF of China (71461027, 11661084), Innovative talent team in Guizhou Province (Qian Ke He Pingtai Rencai[2016]5619), New academic talents and innovative exploration fostering project(Qian Ke He Pingtai Rencai[2017]5727-21), Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2020]094)

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Abstract

In this paper, let $ \mathcal{A} = (a_{i_1 i_2 \cdots i_m } )\in{\mathbb{R}}^{[m, n]} $, when $ m\geq 4 $, based on the condition $ ||x||_2 = 1 $, a new $ Z $-eigenvalue localization set for tensors is given. And we extend the Geršhgorin-type localization set for Z-eigenvalues of fourth order tensors to higher order tensors. As an application, a sharper upper bound for the Z-spectral radius of nonnegative tensors is obtained. Let $ \mathcal{H} $ be a $ k $-uniform hypergraph with $ k\geq 4 $ and $ \mathcal{A}(\mathcal{H}) $ be the adjacency tensor of $ \mathcal{H} $, a new upper bound for the Z-spectral radius $ \rho (\mathcal{H}) $ is also presented. Finally, a checkable sufficient condition for the positive definiteness of even-order tensors and asymptotically stability of time-invariant polynomial systems is also given.

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Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 65F15, 65F10.

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Figure 1. Comparisons of Z-eigenvalue inclusion sets

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Figure 2. Comparisons of Geršhgorin-type Z-eigenvalue inclusion sets

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Table 1. Comparisons with the existed upper bounds

	Example 2	Example 3
$ \rho (\mathcal{A}) $	3.1092	7.3525
Bound (2)	5.3333	40
Bound (3)	5.0437	25
Bound (4)	5.2846	-
Bound (5)	5.1935	-
Theorem 4.2 of [12]	4.4632	-
Theorem 3.1	4	$ 20 $

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