Article Contents
Article Contents

# $E$-eigenvalue localization sets for tensors

• * 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)

• 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.

 Citation:

• 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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