Research

16 pages, 298 KiB

Open AccessArticle

H

-Tensors and the Estimation Inequalities for the Spectral Radius of Nonnegative Tensors

by Xincun Wang and Hongbin Lv

Symmetry 2023, 15(2), 439; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15020439 - 07 Feb 2023

Cited by 1 | Viewed by 1007

Abstract

In this paper, we study two classes of quasi-double diagonally dominant tensors and prove they are

H

-tensors. Numerical examples show that two classes of

H

-tensors are mutually exclusive. Thus, we extend the decision conditions of

H

-tensors. Based on these two [...] Read more.

In this paper, we study two classes of quasi-double diagonally dominant tensors and prove they are

H

-tensors. Numerical examples show that two classes of

H

-tensors are mutually exclusive. Thus, we extend the decision conditions of

H

-tensors. Based on these two classes of tensors, two estimation inequalities for the upper and lower bounds for the spectral radius of nonnegative tensors are obtained. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

7 pages, 282 KiB

Open AccessArticle

Classification of Irreducible

Z_{+}

-Modules of a

Z_{+}

-Ring Using Matrix Equations

by Zhichao Chen and Ruju Zhao

Symmetry 2022, 14(12), 2598; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14122598 - 08 Dec 2022

Cited by 1 | Viewed by 550

Abstract

This paper aims to investigate and categorize all inequivalent and irreducible

Z_{+}

-modules of a commutative unit

Z_{+}

-ring

A

, equipped with set {1, x, y,

x y

} satisfying [...] Read more.

This paper aims to investigate and categorize all inequivalent and irreducible

Z_{+}

-modules of a commutative unit

Z_{+}

-ring

A

, equipped with set {1, x, y,

x y

} satisfying

x^{2} = 1, y^{2} = 1

as a

Z_{+}

-basis by using matrix equations, which was part of a call for a Special Issue about matrix inequalities and equations by Symmetry. If the rank of the

Z_{+}

-module

n \leq 2

, we prove that there are finitely many inequivalent and irreducible

Z_{+}

-modules, respectively, one and three. However, if

n \geq 3

, there is no irreducible

Z_{+}

-module. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

10 pages, 307 KiB

Open AccessArticle

A Shift-Deflation Technique for Computing a Large Quantity of Eigenpairs of the Generalized Eigenvalue Problems

by Wei Wei, Xiaoping Chen, Xueying Shi and An Luo

Symmetry 2022, 14(12), 2547; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14122547 - 02 Dec 2022

Viewed by 975

Abstract

In this paper, we propose a shift-deflation technique for the generalized eigenvalue problems. This technique consists of the following two stages: the shift of converged eigenvalues to zeros, and the deflation of these shifted eigenvalues. By performing the above technique, we construct a [...] Read more.

In this paper, we propose a shift-deflation technique for the generalized eigenvalue problems. This technique consists of the following two stages: the shift of converged eigenvalues to zeros, and the deflation of these shifted eigenvalues. By performing the above technique, we construct a new generalized eigenvalue problem with a lower dimension which shares the same eigenvalues with the original generalized eigenvalue problem except for the converged ones. In addition, we consider the relations of the eigenvectors before and after performing the technique. Finally, numerical experiments show the effectiveness and robustness of the proposed method. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

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12 pages, 293 KiB

Open AccessArticle

Controlling Problem within a Class of Two-Level Positive Maps

by Farrukh Mukhamedov and Izzat Qaralleh

Symmetry 2022, 14(11), 2280; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112280 - 31 Oct 2022

Cited by 1 | Viewed by 810

Abstract

This paper aims to define the set of unital positive maps on

M_{2} (C)

by means of quantum Lotka–Volterra operators which are quantum analogues of the classical Lotka–Volterra operators. Furthermore, a quantum control problem within the class of quantum Lotka–Volterra [...] Read more.

This paper aims to define the set of unital positive maps on

M_{2} (C)

by means of quantum Lotka–Volterra operators which are quantum analogues of the classical Lotka–Volterra operators. Furthermore, a quantum control problem within the class of quantum Lotka–Volterra operators are studied. The proposed approach will lead to the understanding of the behavior of the classical Lotka–Volterra systems within a quantum framework. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

13 pages, 295 KiB

Open AccessArticle

Screw Motion via Matrix Algebra in Three-Dimensional Generalized Space

by Ümit Ziya Savcı

Symmetry 2022, 14(11), 2235; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112235 - 25 Oct 2022

Cited by 1 | Viewed by 1233

Abstract

This paper aims to investigate the screw motion in generalized space. For this purpose, firstly, the change in the screw coordinates is analyzed according to the motion of the reference frames. Moreover, the special cases of this change, such as pure rotation and [...] Read more.

This paper aims to investigate the screw motion in generalized space. For this purpose, firstly, the change in the screw coordinates is analyzed according to the motion of the reference frames. Moreover, the special cases of this change, such as pure rotation and translation, are discussed. Matrix multiplication and the properties of dual numbers are used to obtain dual orthogonal matrices, which are used to simplify the manipulation of screw motion in generalized space. In addition, the dual angular velocity matrix is calculated and shows that the exponential of this matrix can represent the screw displacement in the generalized space. Finally, to support our results, we give two examples of screw motion, the rotation part of which is elliptical and hyperbolic. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

15 pages, 308 KiB

Open AccessArticle

A Note on a Minimal Irreducible Adjustment Pagerank

by Yuehua Feng, Yongxin Dong and Jianxin You

Symmetry 2022, 14(8), 1640; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081640 - 09 Aug 2022

Viewed by 898

Abstract

The stochastic modification and irreducible modification in PageRank produce large web link changes correspondingly. To get a minimal irreducible web link adjustment, a PageRank model of minimal irreducible adjustment and its lumping method are discussed by Li, Chen, and Song. In this paper, [...] Read more.

The stochastic modification and irreducible modification in PageRank produce large web link changes correspondingly. To get a minimal irreducible web link adjustment, a PageRank model of minimal irreducible adjustment and its lumping method are discussed by Li, Chen, and Song. In this paper, we provide alternative proofs for the minimal irreducible PageRank by a new type of similarity transformation matrices. To further provide theorems and fast algorithms on a reduced matrix, an

4 \times 4

block matrix partition case of the minimal irreducible PageRank model is utilized and analyzed. For some real applications of our results, a lumping algorithm used for speeding up PageRank vector computations is also presented. Numerical results are also reported to show the efficiency of the proposed algorithm. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

19 pages, 285 KiB

Open AccessArticle

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

by Zhuo-Heng He, Xiao-Na Zhang, Yun-Fan Zhao and Shao-Wen Yu

Symmetry 2022, 14(6), 1273; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061273 - 20 Jun 2022

Cited by 2 | Viewed by 1066

Abstract

Let

H

be the real quaternion algebra and

H^{m \times n}

denote the set of all

m \times n

matrices over

H

. For

A \in H^{m \times n},

we denote by

A_{ϕ}

the

n \times m

matrix obtained [...] Read more.

Let

H

be the real quaternion algebra and

H^{m \times n}

denote the set of all

m \times n

matrices over

H

. For

A \in H^{m \times n},

we denote by

A_{ϕ}

the

n \times m

matrix obtained by applying

ϕ

entrywise to the transposed matrix

A^{T},

where

ϕ

is a non-standard involution of

H

.

A \in H^{n \times n}

is said to be

ϕ

-skew-Hermicity if

A = - A_{ϕ}

. In this paper, we provide some necessary and sufficient conditions for the existence of a

ϕ

-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns

A_{i} X_{i} {(A_{i})}_{ϕ} + B_{i} X_{i + 1} {(B_{i})}_{ϕ} = C_{i}, (i = 1, 2, 3), A_{4} X_{4} {(A_{4})}_{ϕ} = C_{4}

. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

10 pages, 298 KiB

Open AccessArticle

Spectral Distribution and Numerical Methods for Rational Eigenvalue Problems

by Xiaoping Chen, Wei Wei, Xueying Shi and An Luo

Symmetry 2022, 14(6), 1270; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061270 - 20 Jun 2022

Cited by 1 | Viewed by 1220

Abstract

Rational eigenvalue problems (REPs) have important applications in engineering applications and have attracted more and more attention in recent years. Based on the theory of low-rank modification, we discuss the spectral properties and distribution of the symmetric rational eigenvalue problems, and present two [...] Read more.

Rational eigenvalue problems (REPs) have important applications in engineering applications and have attracted more and more attention in recent years. Based on the theory of low-rank modification, we discuss the spectral properties and distribution of the symmetric rational eigenvalue problems, and present two numerical iteration methods for the above REPs. Numerical experiments demonstrate the effectiveness of our proposed methods. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

18 pages, 336 KiB

Open AccessArticle

Constructing Dixon Matrix for Sparse Polynomial Equations Based on Hybrid and Heuristics Scheme

by Guoqiang Deng, Niuniu Qi, Min Tang and Xuefeng Duan

Symmetry 2022, 14(6), 1174; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061174 - 07 Jun 2022

Cited by 1 | Viewed by 1180

Abstract

Solving polynomial equations inevitably faces many severe challenges, such as easily occupying storage space and demanding prohibitively expensive computation resources. There has been considerable interest in exploiting the sparsity to improve computation efficiency, since asymmetry phenomena are prevalent in scientific and engineering fields, [...] Read more.

Solving polynomial equations inevitably faces many severe challenges, such as easily occupying storage space and demanding prohibitively expensive computation resources. There has been considerable interest in exploiting the sparsity to improve computation efficiency, since asymmetry phenomena are prevalent in scientific and engineering fields, especially as most of the systems in real applications have sparse representations. In this paper, we propose an efficient parallel hybrid algorithm for constructing a Dixon matrix. This approach takes advantage of the asymmetry (i.e., sparsity) in variables of the system and introduces a heuristics strategy. Our method supports parallel computation and has been implemented on a multi-core system. Through time-complexity analysis and extensive benchmarks, we show our new algorithm has significantly reduced computation and memory overhead. In addition, performance evaluation via the Fermat–Torricelli point problem demonstrates its effectiveness in combinatorial geometry optimizations. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

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16 pages, 426 KiB

Open AccessArticle

An Efficient Algorithm for Solving the Matrix Optimization Problem in the Unsupervised Feature Selection

by Chunmei Li and Wen Wu

Symmetry 2022, 14(3), 462; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030462 - 25 Feb 2022

Viewed by 1844

Abstract

In this paper, we consider the symmetric matrix optimization problem arising in the process of unsupervised feature selection. By relaxing the orthogonal constraint, this problem is transformed into a constrained symmetric nonnegative matrix optimization problem, and an efficient algorithm is designed to solve [...] Read more.

In this paper, we consider the symmetric matrix optimization problem arising in the process of unsupervised feature selection. By relaxing the orthogonal constraint, this problem is transformed into a constrained symmetric nonnegative matrix optimization problem, and an efficient algorithm is designed to solve it. The convergence theorem of the new algorithm is derived. Finally, some numerical examples show that the new method is feasible. Notably, some simulation experiments in unsupervised feature selection illustrate that our algorithm is more effective than the existing algorithms. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

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