Advances in Linear Algebra with Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (30 August 2024) | Viewed by 12798

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Departamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
Interests: numerical linear algebra; computer-aided geometric design; approximation theory; numerical analysis
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Special Issue Information

Dear Colleagues,

Linear Algebra is the discipline of mathematics dealing with linear equations, linear maps, and their representations in vector spaces and through matrices. The Gaussian elimination procedure can be considered the birth of the discipline. Systems of linear equations were related to what is now called Cartesian geometry. In this sense, lines and planes are represented by linear equations, and their intersections can be computed by solving systems of linear equations. Linear algebra has undergone  great development since its beginnings, extending its applications beyond Cartesian geometry.

Nowadays, Linear Algebra has a lot of applications in all the sciences and in most of the engineering branches. In particular, they are used to deal with important problems in statistics, such as Markov Chains, in the deterministic and stochastic modelling of systems, in Economics, in Data Science, in Signal Processing, in Mathematical Biology, in Graph Theory, and in many others.

In turn, Multilinear Algebra extends the methods of linear algebra. The paper of matrices in linear algebra is played by tensors in multilinear algebra. It also has many applications in many different fields.

The main purpose of this Special Issue of Axioms is to gather recent results with applications of linear and multilinear algebra. In this sense, the application of methods of numerical linear algebra in Science, Engineering, and Economics are welcome.

We cordially invite you to present your recent contributions to this Special Issue.

Prof. Dr. Jorge Delgado Gracia
Guest Editor

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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. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

  • matrix theory
  • tensor theory
  • numerical methods in linear algebra
  • structured matrices and tensors
  • direct and iterative solution methods
  • accuracy methods in linear algebra
  • applications of linear algebra

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Related Special Issue

Published Papers (12 papers)

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Research

24 pages, 288 KiB  
Article
Some New Algebraic Method Developments in the Characterization of Matrix Equalities
by Yongge Tian
Axioms 2024, 13(10), 657; https://doi.org/10.3390/axioms13100657 - 24 Sep 2024
Viewed by 466
Abstract
Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and [...] Read more.
Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
14 pages, 348 KiB  
Article
On Some Properties for Cofiniteness of Submonoids and Ideals of an Affine Semigroup
by Carmelo Cisto
Axioms 2024, 13(7), 488; https://doi.org/10.3390/axioms13070488 - 20 Jul 2024
Viewed by 706
Abstract
Let S and C be affine semigroups in Nd such that SC. We provide a characterization for the set CS to be finite, together with a procedure and computational tools to check whether such a set is [...] Read more.
Let S and C be affine semigroups in Nd such that SC. We provide a characterization for the set CS to be finite, together with a procedure and computational tools to check whether such a set is finite and, if so, compute its elements. As a consequence of this result, we provide a characterization for an ideal I of an affine semigroup S so that SI is a finite set. If so, we provide some procedures to compute the set SI. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
28 pages, 400 KiB  
Article
On Semi-Vector Spaces and Semi-Algebras with Applications in Fuzzy Automata
by Giuliano G. La Guardia, Jocemar Q. Chagas, Ervin K. Lenzi, Leonardo Pires, Nicolás Zumelzu and Benjamín Bedregal
Axioms 2024, 13(5), 308; https://doi.org/10.3390/axioms13050308 - 8 May 2024
Viewed by 897
Abstract
In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R0+. More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues [...] Read more.
In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R0+. More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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9 pages, 225 KiB  
Article
Conditions When the Problems of Linear Programming Are Algorithmically Unsolvable
by Viktor Chernov and Vladimir Chernov
Axioms 2024, 13(5), 293; https://doi.org/10.3390/axioms13050293 - 27 Apr 2024
Viewed by 800
Abstract
We study the properties of the constructive linear programming problems. The parameters of linear functions in such problems are constructive real numbers. Solving such a problem involves finding the optimal plan with the constructive real number components. We show that it is impossible [...] Read more.
We study the properties of the constructive linear programming problems. The parameters of linear functions in such problems are constructive real numbers. Solving such a problem involves finding the optimal plan with the constructive real number components. We show that it is impossible to have an algorithm that solves an arbitrary constructive real programming problem. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
17 pages, 377 KiB  
Article
The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra
by José-Javier Martínez
Axioms 2024, 13(4), 258; https://doi.org/10.3390/axioms13040258 - 14 Apr 2024
Viewed by 943
Abstract
The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 [...] Read more.
The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
18 pages, 288 KiB  
Article
A Heuristic Method for Solving Polynomial Matrix Equations
by Juan Luis González-Santander and Fernando Sánchez Lasheras
Axioms 2024, 13(4), 239; https://doi.org/10.3390/axioms13040239 - 4 Apr 2024
Viewed by 855
Abstract
We propose a heuristic method to solve polynomial matrix equations of the type k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The [...] Read more.
We propose a heuristic method to solve polynomial matrix equations of the type k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
23 pages, 404 KiB  
Article
Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel
by Raquel Pinto, Marcos Spreafico and Carlos Vela
Axioms 2024, 13(3), 197; https://doi.org/10.3390/axioms13030197 - 15 Mar 2024
Cited by 1 | Viewed by 1076
Abstract
In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We [...] Read more.
In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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16 pages, 271 KiB  
Article
C-S and Strongly C-S Orthogonal Matrices
by Xiaoji Liu, Ying Liu and Hongwei Jin
Axioms 2024, 13(2), 110; https://doi.org/10.3390/axioms13020110 - 5 Feb 2024
Viewed by 1078
Abstract
In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and [...] Read more.
In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
16 pages, 305 KiB  
Article
T-BT Inverse and T-GC Partial Order via the T-Product
by Hongxing Wang and Wei Wen
Axioms 2023, 12(10), 929; https://doi.org/10.3390/axioms12100929 - 28 Sep 2023
Viewed by 703
Abstract
In this paper, we extend the BT inverse to the set of third-order tensors, and we call it the T-BT inverse. We give characterizations and properties of the inverse by applying tensor decomposition. Based on the inverse, we introduce a new binary relation: [...] Read more.
In this paper, we extend the BT inverse to the set of third-order tensors, and we call it the T-BT inverse. We give characterizations and properties of the inverse by applying tensor decomposition. Based on the inverse, we introduce a new binary relation: T-BT order. Furthermore, by applying the T-BT order, we introduce a generalized core partial order (called T-GC partial order). Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
17 pages, 616 KiB  
Article
On the Total Positivity and Accurate Computations of r-Bell Polynomial Bases
by Esmeralda Mainar, Juan Manuel Peña and Beatriz Rubio
Axioms 2023, 12(9), 839; https://doi.org/10.3390/axioms12090839 - 29 Aug 2023
Viewed by 904
Abstract
A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy [...] Read more.
A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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16 pages, 618 KiB  
Article
Green Matrices, Minors and Hadamard Products
by Jorge Delgado, Guillermo Peña and Juan Manuel Peña
Axioms 2023, 12(8), 774; https://doi.org/10.3390/axioms12080774 - 10 Aug 2023
Cited by 2 | Viewed by 1370
Abstract
Green matrices are interpreted as discrete version of Green functions and are used when working with inhomogeneous linear system of differential equations. This paper discusses accurate algebraic computations using a recent procedure to achieve an important factorization of these matrices with high relative [...] Read more.
Green matrices are interpreted as discrete version of Green functions and are used when working with inhomogeneous linear system of differential equations. This paper discusses accurate algebraic computations using a recent procedure to achieve an important factorization of these matrices with high relative accuracy and using alternative accurate methods. An algorithm to compute any minor of a Green matrix with high relative accuracy is also presented. The bidiagonal decomposition of the Hadamard product of Green matrices is obtained. Illustrative numerical examples are included. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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12 pages, 804 KiB  
Article
Tensor Eigenvalue and SVD from the Viewpoint of Linear Transformation
by Xinzhu Zhao, Bo Dong, Bo Yu and Yan Yu
Axioms 2023, 12(5), 485; https://doi.org/10.3390/axioms12050485 - 17 May 2023
Viewed by 1480
Abstract
A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization [...] Read more.
A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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