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Optimal $ Z $-eigenvalue inclusion intervals of tensors and their applications

  • * Corresponding author: Zhen Chen

    * Corresponding author: Zhen Chen
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  • Firstly, a weakness of Theorem 3.2 in [Journal of Industrial and Management Optimization, 17(2) (2021) 687-693] is pointed out. Secondly, a new Geršgorin-type $ Z $-eigenvalue inclusion interval for tensors is given. Subsequently, another Geršgorin-type $ Z $-eigenvalue inclusion interval with parameters for even order tensors is presented. Thirdly, by selecting appropriate parameters some optimal intervals are provided and proved to be tighter than some existing results. Finally, as an application, some sufficient conditions for the positive definiteness of homogeneous polynomial forms as well as the asymptotically stability of time-invariant polynomial systems are obtained. As another application, bounds of $ Z $-spectral radius of weakly symmetric nonnegative tensors are presented, which are used to estimate the convergence rate of the greedy rank-one update algorithm and derive bounds of the geometric measure of entanglement of symmetric pure state with nonnegative amplitudes.

    Mathematics Subject Classification: Primary: 15A18, 15A42; Secondary: 15A69.

    Citation:

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

    Method $ \varrho(\mathcal{A})\leq $
    Theorem 5.5, i.e., Corollary 4.5 of [39] 26.0000
    Theorem 3.3 of [25] 25.7771
    Theorem 3.4 of [47], where $ S=\{1\},\bar{S}=\{2\} $ 25.7382
    Theorem 4.5 of [41] 25.7382
    Theorem 3.5 of [10] 25.6437
    Theorem 6 of [11] 25.6437
    Theorem 4 of [43], where $ S=\{1\},\bar{S}=\{2\} $ 25.6437
    Theorem 7 of [35] 25.4807
    Theorem 7 of [44] 25.4807
    Theorem 2.9 of [27] 23.8617
    Theorem 5 of [46] 22.5426
    Theorem 3.1 of [36] 21.8172
    Theorem 5.6 16.0000
    Corollary 9 14.5000
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