Further Study on Z-Eigenvalue Localization Set and Positive Definiteness of Fourth-Order Tensors

Sun, Linxuan; Wang, Gang; Liu, Lixia

doi:10.1007/s40840-020-00939-2

Bull. Malays. Math. Sci. Soc.

2020

DOI: 10.1007/s40840-020-00939-2

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Further Study on Z-Eigenvalue Localization Set and Positive Definiteness of Fourth-Order Tensors

Linxuan Sun

Gang Wang

Lixia Liu

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Cited by 14 publications

(2 citation statements)

References 25 publications

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“…In this section, we give a new Z-eigenvalue localization set for tensors with m ≥ 4. We need the following lemma [19].…”

mentioning

confidence: 99%

New Z-eigenvalue localization sets for tensors with applications

Liu

2022

JIMO

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In this paper, let A = (a i 1 i 2 •••im ) ∈ R [m,n] , when m ≥ 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 H be a k-uniform hypergraph with k ≥ 4 and A(H) be the adjacency tensor of H, a new upper bound for the Z-spectral radius ρ(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.

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“…In this section, we give a new Z-eigenvalue localization set for tensors with m ≥ 4. We need the following lemma [19].…”

mentioning

confidence: 99%

New Z-eigenvalue localization sets for tensors with applications

Liu

2022

JIMO

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“…Besides that the bounds can be used to estimate the geometric measure of entanglement for symmetric pure states with nonnegative amplitudes in quantum mechanics [44]. The related results are showed in [10,11,12,25,27,35,37,38,39,40,41,43,46,47,48].…”

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confidence: 99%

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

Sang

Chen

2022

JIMO

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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.

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Optimal $Z$-Eigenvalue Inclusion Intervals for Even Order Tensors and Their Applications

Zhao

2021

Acta Appl Math

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Further Study on Z-Eigenvalue Localization Set and Positive Definiteness of Fourth-Order Tensors

Cited by 14 publications

References 25 publications

New Z-eigenvalue localization sets for tensors with applications

New Z-eigenvalue localization sets for tensors with applications

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

Optimal $Z$-Eigenvalue Inclusion Intervals for Even Order Tensors and Their Applications

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