Axioms

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9 pages, 224 KiB

Open AccessArticle

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

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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

Open AccessArticle

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 338

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

Open AccessArticle

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 - 04 Apr 2024

Viewed by 466

Abstract

We propose a heuristic method to solve polynomial matrix equations of the type

\sum_{k = 1}^{m} a_{k} X^{k} = B

, where

a_{k}

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

\sum_{k = 1}^{m} a_{k} X^{k} = B

, where

a_{k}

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

Open AccessArticle

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

Viewed by 697

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

Open AccessArticle

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 - 05 Feb 2024

Viewed by 786

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

A^{S} B = 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

A^{S} B = 0

and

B A^{S} = 0

, where

A^{S}

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

Open AccessArticle

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 492

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

Open AccessFeature PaperArticle

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 645

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

Open AccessArticle

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 1 | Viewed by 1045

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

Open AccessArticle

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 1087

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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