$ E $-eigenvalue localization sets for tensors

Caili Sang; Zhen Chen

doi:10.3934/jimo.2019042

Article Contents

2020, Volume 16, Issue 4: 2045-2063. Doi: 10.3934/jimo.2019042

This issue Previous Article Robust sensitivity analysis for linear programming with ellipsoidal perturbation Next Article

$ E $-eigenvalue localization sets for tensors

Caili Sang^1,2, and
Zhen Chen^1, ,

1.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
2.
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

^* Corresponding author: Zhen Chen
^* Corresponding author: Zhen Chen

Received: August 31, 2018

Revised: November 30, 2018

Early access: May 2019

Published: May 2020

This work is supported by National Natural Science Foundation of China (No. 11501141), Science and Technology Projects of Education Department of Guizhou Province (Grant No. KY[2015]352), and Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066)

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Abstract

Several existing $Z$-eigenvalue localization sets for tensors are first generalized to $E$-eigenvalue localization sets. And then two tighter $ E$-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the $ Z$-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

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Mathematics Subject Classification: Primary: 15A18, 15A69, 15A72, 15A42.

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Figure 1. Comparisons of $ \mathcal{K}(\mathcal{A}) $, $ \mathcal{L}(\mathcal{A}) $, $ \Psi(\mathcal{A}) $, $ \Upsilon(\mathcal{A}) $ and $ \Omega(\mathcal{A}) $.

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Figure 2. Comparisons of $ \Omega(\mathcal{A}) $ and $ \triangle(\mathcal{A}) $.

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Table 1. Upper bounds of $ \varrho(\mathcal{A}) $

Method	$ \varrho(\mathcal{A})\leq $
Theorem 3.1, i.e., Corollary 4.5 of [29]	23.0000
Theorem 3.3 of [16]	22.8625
Theorem 3.4 of [35], where $ S=\{3\},\bar{S}=\{1,2\} $	22.8521
Theorem 3.2, i.e., Theorem 4.5 of [31]	22.8149
Theorem 4.7 of [31]	22.7759
Theorem 2.9 of [23]	22.7217
Theorem 3.5 of [9]	22.7163
Theorem 4.6 of [31]	22.6478
Theorem 6 of [10]	22.6290
Theorem 3.3, i.e., Theorem 5 of [34]	22.5000
Theorem 4 of [30], where $ S=\{3\},\bar{S}=\{1,2\} $	22.4195
Theorem 3.4, i.e., Theorem 7 of [28]	22.2122
Theorem 3.5	21.2604

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