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$ E $-eigenvalue localization sets for tensors

  • * Corresponding author: Zhen Chen

    * Corresponding author: Zhen Chen

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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  • 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.

    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}) $.

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

    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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