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Symmetry
Special Issues
Tensors and Matrices in Symmetry with Applications
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Tensors and Matrices in Symmetry with Applications

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

Deadline for manuscript submissions: closed (10 April 2023) | Viewed by 7923

Special Issue Editors

Prof. Dr. Wen Li

E-Mail Website1 Website2
Guest Editor
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Interests: tensor and matrix computations
Dr. Yannan Chen

E-Mail Website1 Website2
Guest Editor
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Interests: tensor computation and optimization
Prof. Dr. Zhigang Jia

E-Mail Website1 Website2
Guest Editor
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
Interests: quaternion matrix theory; image processing; machine learning; scientific computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Matrices and tensors are shown to be foundational and powerful tools for capturing linear and multilinear interactions in various disciplines and have extensive applications in science and engineering. Symmetry plays important roles in models, theory, and algorithms of matrices and tensors. Tensor decompositions, tensor spectral theory, tensor computations, numerical linear and multilinear algebra, structure matrices, tensor (matrix) low rank approximations, and their applications have been active research areas in recent years.

Although a remarkable number of papers have been contributed to theory, algorithms, and applications of tensors and matrices, it is still challenging and particularly important to develop problem-driven models, structure-exploiting algorithms, and associated mathematical theory for tensor- and matrix-related problems. The goal of this Special Issue is to attract original research papers on the models, theory, algorithms, and applications of tensors and matrices. These applications include automatic control, statistical inference, machine learning, data recovery, computer vision, image and video processing, graph and network analysis, and other data-driven applications.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Tensors and Matrices in Symmetry with Applications” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Prof. Dr. Wen Li
Dr. Yannan Chen
Prof. Dr. Zhigang Jia
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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

  • tensor decompositions
  • tensor spectral theory
  • tensor computations
  • numerical linear and multilinear algebra with applications
  • structure matrix analysis in applications
  • tensor (matrix) low rank approximations in data analysis

Published Papers (6 papers)

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Research

10 pages, 836 KiB  
Article
Solving the Adaptive Cubic Regularization Sub-Problem Using the Lanczos Method
by Zhi Zhu and Jingya Chang
Symmetry 2022, 14(10), 2191; https://doi.org/10.3390/sym14102191 - 18 Oct 2022
Viewed by 1147
Abstract
The adaptive cubic regularization method solves an unconstrained optimization model by using a three-order regularization term to approximate the objective function at each iteration. Similar to the trust-region method, the calculation of the sub-problem highly affects the computing efficiency. The Lanczos method is [...] Read more.
The adaptive cubic regularization method solves an unconstrained optimization model by using a three-order regularization term to approximate the objective function at each iteration. Similar to the trust-region method, the calculation of the sub-problem highly affects the computing efficiency. The Lanczos method is an useful tool for simplifying the objective function in the sub-problem. In this paper, we implement the adaptive cubic regularization method with the aid of the Lanczos method, and analyze the error of Lanczos approximation. We show that both the error between the Lanczos objective function and the original cubic term, and the error between the solution of the Lanczos approximation and the solution of the original cubic sub-problem are bounded up by the condition number of the optimal Hessian matrix. Furthermore, we compare the numerical performances of the adaptive cubic regularization algorithm when using the Lanczos approximation method and the adaptive cubic regularization algorithm without using the Lanczos approximation for unconstrained optimization problems. Numerical experiments show that the Lanczos method improves the computation efficiency of the adaptive cubic method remarkably. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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14 pages, 310 KiB  
Article
A Two-Step Iteration Method for Vertical Linear Complementarity Problems
by Yunlin Song, Hua Zheng, Xiaoping Lu and Seak-Weng Vong
Symmetry 2022, 14(9), 1882; https://doi.org/10.3390/sym14091882 - 08 Sep 2022
Cited by 4 | Viewed by 1123
Abstract
In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the [...] Read more.
In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the parameter matrix compared to the recent results. Numerical examples show that the proposed method is efficient with the two-step technique and confirm the improvement of the theoretical results. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
6 pages, 251 KiB  
Article
Finding Solutions to the Yang–Baxter-like Matrix Equation for Diagonalizable Coefficient Matrix
by Dongmei Chen and Xuerong Yong
Symmetry 2022, 14(8), 1577; https://doi.org/10.3390/sym14081577 - 31 Jul 2022
Cited by 2 | Viewed by 1075
Abstract
Let A be a diagonalizable complex matrix. In this paper, we discuss finding solutions to the Yang–Baxter-like matrix equation AXA=XAX. We then present a concrete example to illustrate the validity of the results obtained. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
7 pages, 310 KiB  
Article
An Improved Convergence Theorem of the Newton-Based AOR Method for Generalized Absolute Value Equations
by Raojie Chen, Xiaofei Peng and Wensong Yu
Symmetry 2022, 14(6), 1249; https://doi.org/10.3390/sym14061249 - 16 Jun 2022
Viewed by 1199
Abstract
For solving the large sparse generalized absolute value equations, recently a Newton-based accelerated over-relaxation (NAOR) method was investigated. In this paper, we widen the convergence regions for the parameters and establish a new convergence theorem of the NAOR method when the system matrix [...] Read more.
For solving the large sparse generalized absolute value equations, recently a Newton-based accelerated over-relaxation (NAOR) method was investigated. In this paper, we widen the convergence regions for the parameters and establish a new convergence theorem of the NAOR method when the system matrix is an H+-matrix. Numerical examples demonstrate that the NAOR method has a better convergence performance when the parameters are taken according to the proposed convergence theorem. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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14 pages, 339 KiB  
Article
An Efficient Method for Split Quaternion Matrix Equation X − Af(X)B = C
by Shufang Yue, Ying Li, Anli Wei and Jianli Zhao
Symmetry 2022, 14(6), 1158; https://doi.org/10.3390/sym14061158 - 04 Jun 2022
Cited by 1 | Viewed by 1279
Abstract
In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH} [...] Read more.
In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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14 pages, 317 KiB  
Article
A New Method of Solving Special Solutions of Quaternion Generalized Lyapunov Matrix Equation
by Zhihong Liu, Ying Li, Xueling Fan and Wenxv Ding
Symmetry 2022, 14(6), 1120; https://doi.org/10.3390/sym14061120 - 29 May 2022
Cited by 1 | Viewed by 1205
Abstract
In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed [...] Read more.
In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method. Full article
(This article belongs to the Special Issue Tensors and Matrices in Symmetry with Applications)
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