Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 13290

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Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: associative rings and algebras; combinatorics; information and communication; linear and multilinear algebra; matrix theory; numerical analysis; operations research; mathematical programming; operator theory; quantum theory; statistics
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Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: matrix theory; quaternion algebra; numerical linear algebra
Special Issues, Collections and Topics in MDPI journals

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School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, China
Interests: numerical linear and nonlinear algebra; image information security; big data analysis and computation; structured low rank approximation and its applications
School of Mathematics and Statistics, Hainan Normal University, Haikou, China
Interests: matrix analysis; matrix inequalities; mathematical inequalities; numerical algebra

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Guest Editor
School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China
Interests: pure mathematics

Special Issue Information

Dear Colleagues,

The 2021 International Conference on Matrix Inequalities and Matrix Equations (MIME2021) took place in Hainan, China, on 25–28 November 2021.

The purpose of the conference is to stimulate research and foster interaction of researchers interested in matrix inequalities, matrix equations, and their applications. Hopefully, the informal conference atmosphere could ensure the exchange of ideas from different research areas.

Topics to be covered include (but are not limited to) all the research areas of tensor and matrix equations, tensor and matrix inequalities and their applications.

For more information, please click on the following link: https://math.shu.edu.cn/info/1043/3256.htm

Prof. Dr. Qing-Wen Wang
Dr. Zhuo-Heng He
Dr. Xuefeng Duan
Dr. Xiao-Hui Fu
Dr. Guang-Jing Song
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • matrix equation
  • matrix inequalities
  • generalized inverses
  • tensor equation
  • algorithm

Published Papers (10 papers)

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Research

16 pages, 298 KiB  
Article
Quasi-Double Diagonally Dominant 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://doi.org/10.3390/sym15020439 - 7 Feb 2023
Cited by 1 | Viewed by 1086
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
7 pages, 282 KiB  
Article
Classification of Irreducible Z+-Modules of a Z+-Ring Using Matrix Equations
by Zhichao Chen and Ruju Zhao
Symmetry 2022, 14(12), 2598; https://doi.org/10.3390/sym14122598 - 8 Dec 2022
Cited by 1 | Viewed by 606
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, xy} 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, xy} satisfying x2=1,y2=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 n2, we prove that there are finitely many inequivalent and irreducible Z+-modules, respectively, one and three. However, if n3, there is no irreducible Z+-module. Full article
10 pages, 307 KiB  
Article
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://doi.org/10.3390/sym14122547 - 2 Dec 2022
Viewed by 1054
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
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12 pages, 293 KiB  
Article
Controlling Problem within a Class of Two-Level Positive Maps
by Farrukh Mukhamedov and Izzat Qaralleh
Symmetry 2022, 14(11), 2280; https://doi.org/10.3390/sym14112280 - 31 Oct 2022
Cited by 1 | Viewed by 878
Abstract
This paper aims to define the set of unital positive maps on M2(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 M2(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
13 pages, 295 KiB  
Article
Screw Motion via Matrix Algebra in Three-Dimensional Generalized Space
by Ümit Ziya Savcı
Symmetry 2022, 14(11), 2235; https://doi.org/10.3390/sym14112235 - 25 Oct 2022
Cited by 1 | Viewed by 1381
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
15 pages, 308 KiB  
Article
A Note on a Minimal Irreducible Adjustment Pagerank
by Yuehua Feng, Yongxin Dong and Jianxin You
Symmetry 2022, 14(8), 1640; https://doi.org/10.3390/sym14081640 - 9 Aug 2022
Viewed by 988
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 4×4 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
19 pages, 285 KiB  
Article
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://doi.org/10.3390/sym14061273 - 20 Jun 2022
Cited by 2 | Viewed by 1146
Abstract
Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For AHm×n, we denote by Aϕ the n×m matrix obtained [...] Read more.
Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For AHm×n, we denote by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a non-standard involution of H. AHn×n is said to be ϕ-skew-Hermicity if A=Aϕ. 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 AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4. Full article
10 pages, 298 KiB  
Article
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://doi.org/10.3390/sym14061270 - 20 Jun 2022
Cited by 1 | Viewed by 1310
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
18 pages, 336 KiB  
Article
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://doi.org/10.3390/sym14061174 - 7 Jun 2022
Cited by 1 | Viewed by 1265
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
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16 pages, 426 KiB  
Article
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://doi.org/10.3390/sym14030462 - 25 Feb 2022
Viewed by 1959
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
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