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

Caili Sang; Zhen Chen

doi:10.3934/jimo.2021075

Article Contents

2022, Volume 18, Issue 4: 2435-2468. Doi: 10.3934/jimo.2021075

This issue Previous Article Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system Next Article A new gradient computational formula for optimal control problems with time-delay

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

Caili Sang^1,2, and
Zhen Chen^1, ,

1.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
2.
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

^* Corresponding author: Zhen Chen
^* Corresponding author: Zhen Chen

Received: September 30, 2020

Revised: December 31, 2020

Published: July 2022

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Abstract

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

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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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