scholarly journals Two new eigenvalue localization sets for tensors and theirs applications

2017 ◽  
Vol 15 (1) ◽  
pp. 1267-1276 ◽  
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Linear Algebra ◽  
Symmetric Tensor ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Even Order ◽  
Weakly Symmetric ◽  
Localization Set ◽  
Theoretical Results ◽  
Localization Sets

Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

scholarly journals New Sufficient Condition for the Positive Definiteness of Fourth Order Tensors

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 303 ◽  
Author(s):  
Jun He ◽  
Yanmin Liu ◽  
Junkang Tian ◽  
Zhuanzhou Zhang
Keyword(s):  
Spectral Radius ◽  
Fourth Order ◽  
Upper Bound ◽  
Positive Definiteness ◽  
Sufficient Condition ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Weakly Symmetric ◽  
Localization Set

In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

scholarly journals Improved Brauer-type eigenvalue localization sets for tensors with their applications

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4607-4625
Author(s):  
Zhengge Huang ◽  
Jingjing Cui
Keyword(s):  
Linear Algebra ◽  
Multilinear Algebra ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
New Criteria ◽  
Localization Sets ◽  
Better Than

In this paper, by excluding some sets from the Brauer-type eigenvalue inclusion sets for tensors developed by Bu et al. (Linear Algebra Appl. 512 (2017) 234-248) and Li et al. (Linear and Multilinear Algebra 64 (2016) 727-736), some improved Brauer-type eigenvalue localization sets for tensors are given, which are proved to be much tighter than those put forward by Bu et al. and Li et al. As applications, some new criteria for identifying the nonsingularity of tensors are developed, which are better than some previous results. This fact is illustrated by some numerical examples.

scholarly journals A modified S-type eigenvalue localization set of tensors applications

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6395-6416
Author(s):  
Zhengge Huang ◽  
Ligong Wang ◽  
Zhong Xu ◽  
Jingjing Cui
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Nonnegative Tensors ◽  
Eigenvalue Localization ◽  
Localization Set ◽  
Weakly Irreducible ◽  
Eigenvalues Of Tensors

Based on the S-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors, a modified S-type eigenvalue localization set for tensors is established in this paper by excluding some sets from the existing S-type eigenvalue localization set developed by Huang et al. (arXiv: 1602.07568v1, 2016). The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al. and Huang et al. As its applications, a criteria, which can be utilized for identifying the nonsingularity of tensors, is developed. In addition, we provide new upper and lower bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of weakly irreducible strong M-tensors. These bounds are superior to some previous results, which is illustrated by some numerical examples.

scholarly journals Some new inclusion sets for eigenvalues of tensors with application

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3899-3916
Author(s):  
Zhengge Huang ◽  
Ligong Wang ◽  
Zhong Xu ◽  
Jingjing Cui
Keyword(s):  
Symmetric Tensors ◽  
Eigenvalue Localization ◽  
Even Order ◽  
Localization Sets ◽  
Eigenvalues Of Tensors

In this paper, we are concerned with the eigenvalue inclusion sets for tensors. Some new S-type eigenvalue localization sets for tensors are employed by dividing N = {1,2,..., n} into disjoint subsets S and its complement. Our new sets, are proved to be tighter than that newly derived by Huang et al. (J. Inequal. Appl. 2016 (2016) 254). As applications, we can apply the proposed sets for determining the positive (semi-)definiteness of even-order symmetric tensors. Some examples are given to show the sharpness of our new sets in contrast with the known ones, and verify the effectiveness of those in identifying the positive (semi-)definiteness of tensors.

scholarly journals New iterative codes for 𝓗-tensors and an application

2016 ◽  
Vol 14 (1) ◽  
pp. 212-220 ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Polynomial Form ◽  
Numerical Examples ◽  
Even Order

AbstractNew iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals Some improvements on the Ky Fan theorem for tensors

2021 ◽  
Vol 40 (2) ◽  
Author(s):  
Mohsen Tourang ◽  
Mostafa Zangiabadi
Keyword(s):  
Multilinear Algebra ◽  
Numerical Examples ◽  
Eigenvalue Localization ◽  
Fan Theorem ◽  
Linear Multilinear Algebra ◽  
Numer Math ◽  
Ky Fan ◽  
Localization Sets ◽  
Ky Fan Theorem ◽  
Perron Vector

AbstractThe improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

scholarly journals New criteria for H-tensors and an application

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Feng Wang ◽  
Deshu Sun
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Numerical Examples ◽  
Even Order ◽  
New Criteria

AbstractSome new criteria for identifying H-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of results obtained are illustrated by numerical examples.

scholarly journals Practical Criteria for H-Tensors and Their Application

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 155
Author(s):  
Min Li ◽  
Haifeng Sang ◽  
Panpan Liu ◽  
Guorui Huang
Keyword(s):  
Sufficient Conditions ◽  
Symmetric Tensor ◽  
Positive Definiteness ◽  
Tensor Analysis ◽  
Symmetric Tensors ◽  
Numerical Examples ◽  
Even Order

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

scholarly journals E-eigenvalue inclusion theorems for tensors

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3883-3891
Author(s):  
Caili Sang ◽  
Jianxing Zhao
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Inclusion Theorem ◽  
Numerical Examples ◽  
Nonnegative Tensors ◽  
Weakly Symmetric ◽  
Theoretical Results

Two Z-eigenvalue inclusion theorems for tensors presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1): 187-198) are first generalized to E-eigenvalue inclusion theorems. And then a tighter E-eigenvalue inclusion theorem for tensors is established. Based on the new set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

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