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New Z-eigenvalue localization sets for tensors with applications

  • * Corresponding author: Jun He

    * Corresponding author: Jun He 

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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  • 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.

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


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  • Figure 1.  Comparisons of Z-eigenvalue inclusion sets

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

    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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