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Symmetry
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Symmetry in Numerical Linear and Multilinear Algebra
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Symmetry in Numerical Linear and Multilinear Algebra

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

Deadline for manuscript submissions: closed (15 March 2020) | Viewed by 16406

Special Issue Editor

Prof. Dr. Dario Fasino

E-Mail Website
Guest Editor
Department of Mathematics, Computer Science and Physics, University of Udine, 33100 Udine, Italy
Interests: numerical linear algebra; complex networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry is one of the most pervading concepts in numerical linear and multilinear algebra. Mathematical models of real-world phenomena often exhibit algebraic, geometric or combinatorial symmetries, which are encoded into specially structured matrices and tensors. In fact, an increasing number of physical models and data analysis problems involve the manipulation of numerical arrays of which the elements are addressed by two or more indices and possess invariance properties with respect to permutations or shifting of their indices. Exploiting that structure is of paramount importance not only for theoretical analysis but also in order to devise fast and accurate computational cores for, e.g., direct or iterative solution of linear equations, matrix preconditioning and decomposition, eigenvalue/eigenvector computations, solution of matrix equations, etc.  

This Special Issue invites significant and original contributions in numerical linear and multilinear algebra involving symmetry, in a broad sense. Contributions may address theoretical aspects, applications, and related computational issues. We welcome manuscripts addressing matrices and tensors having symmetric or displacement structures, their spectral and computational properties, as well as their occurrence and applications in linear differential equations, inverse problems, signal processing, optimization problems, machine learning, and network science.

Prof. Dario Fasino
Guest Editor

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

  • Special matrices
  • Linear systems
  • Eigenvalues, singular values
  • eigenvectors
  • Matrix equations
  • Error analysis

Published Papers (6 papers)

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Research

11 pages, 1085 KiB  
Article
On Spectral Properties of Doubly Stochastic Matrices
by Mutti-Ur Rehman, Jehad Alzabut, Javed Hussain Brohi and Arfan Hyder
Symmetry 2020, 12(3), 369; https://doi.org/10.3390/sym12030369 - 02 Mar 2020
Cited by 3 | Viewed by 6002
Abstract
The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic [...] Read more.
The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed. Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
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13 pages, 780 KiB  
Article
An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System
by Mubasher Umer, Umar Hayat, Fazal Abbas, Anurag Agarwal and Petko Kitanov
Symmetry 2020, 12(2), 311; https://doi.org/10.3390/sym12020311 - 21 Feb 2020
Cited by 3 | Viewed by 2095
Abstract
In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the [...] Read more.
In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the algorithm using some examples. The proposed algorithm is implemented in MATLAB, using max-plus algebra toolbox. Computationally speaking, our algorithm has a clear advantage over the power algorithm presented by Subiono and van der Woude. Because our algorithm takes 0 . 088783 sec to solve the eigenvalue problem for Latin square presented in Example 2, while the compared one takes 1 . 718662 sec for the same problem. Furthermore, a time complexity comparison is presented, which reveals that the proposed algorithm is less time consuming when compared with some of the existing algorithms. Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
16 pages, 273 KiB  
Article
General Solutions for Descriptor Systems of Coupled Generalized Sylvester Matrix Fractional Differential Equations via Canonical Forms
by Kanjanaporn Tansri and Pattrawut Chansangiam
Symmetry 2020, 12(2), 283; https://doi.org/10.3390/sym12020283 - 14 Feb 2020
Cited by 2 | Viewed by 1556
Abstract
We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a [...] Read more.
We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a matrix canonical form concerning ranks, and matrix partitioning. The procedure aims to reduce the descriptor system to a descriptor system of fractional differential equations. We also impose a condition on coefficient matrices, related to the symmetry of the solution for descriptor systems. It follows that an explicit form of its general solution is given in terms of matrix power series concerning Mittag–Leffler functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases. In particular, we obtain an alternative procedure to solve linear continuous-time descriptor systems via a matrix canonical form. Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
14 pages, 2155 KiB  
Article
A Fast and Exact Greedy Algorithm for the Core–Periphery Problem
by Dario Fasino and Franca Rinaldi
Symmetry 2020, 12(1), 94; https://doi.org/10.3390/sym12010094 - 03 Jan 2020
Cited by 6 | Viewed by 2405
Abstract
The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce [...] Read more.
The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce a well-connected subgraph and share connections with peripheral nodes, while the peripheral nodes are loosely connected to the core nodes and other peripheral nodes. We propose a polynomial-time algorithm to detect core–periphery structures in networks having a symmetric adjacency matrix. The core set is defined as the solution of a combinatorial optimization problem, which has a pleasant symmetry with respect to graph complementation. We provide a complete description of the optimal solutions to that problem and an exact and efficient algorithm to compute them. The proposed approach is extended to networks with loops and oriented edges. Numerical simulations are carried out on both synthetic and real-world networks to demonstrate the effectiveness and practicability of the proposed algorithm. Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
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11 pages, 299 KiB  
Article
Isomorphism Theorems in the Primary Categories of Krasner Hypermodules
by Hossein Shojaei and Dario Fasino
Symmetry 2019, 11(5), 687; https://doi.org/10.3390/sym11050687 - 18 May 2019
Cited by 3 | Viewed by 1723
Abstract
Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of [...] Read more.
Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R-hypermodules with strong single-valued R-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R-homomorphism f : A → B and a ∈ A , we have K e r ( f ) + a = a + K e r ( f ) = { x ∈ A ∣ f ( x ) = f ( a ) } . Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
10 pages, 13152 KiB  
Article
Eventually DSDD Matrices and Eigenvalue Localization
by Caili Sang and Jianxing Zhao
Symmetry 2018, 10(10), 448; https://doi.org/10.3390/sym10100448 - 01 Oct 2018
Cited by 4 | Viewed by 1964
Abstract
Firstly, the relationships among strictly diagonally dominant ( S D D ) matrices, doubly strictly diagonally dominant ( D S D D ) matrices, eventually S D D matrices and eventually D S D D matrices are considered. Secondly, by excluding some proper [...] Read more.
Firstly, the relationships among strictly diagonally dominant ( S D D ) matrices, doubly strictly diagonally dominant ( D S D D ) matrices, eventually S D D matrices and eventually D S D D matrices are considered. Secondly, by excluding some proper subsets of an existing eigenvalue inclusion set for matrices, which do not contain any eigenvalues of matrices, a tighter eigenvalue inclusion set of matrices is derived. As its application, a sufficient condition of determining non-singularity of matrices is obtained. Finally, the infinity norm estimation of the inverse of eventually D S D D matrices is derived. Full article
(This article belongs to the Special Issue Symmetry in Numerical Linear and Multilinear Algebra)
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