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Axioms
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Advances in Linear Algebra
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Advances in Linear Algebra

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

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 6310

Special Issue Editors

Dr. Ivan I. Kyrchei

E-Mail Website
Guest Editor
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Lviv, Ukraine
Interests: linear algebra and its application; algebra of matrices over noncommutative ring; theory of matrices; generalized inverse matrices; matrix and differential matrix equations
Dr. Zhuo-Heng He

E-Mail Website
Co-Guest Editor
Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: matrix theory; quaternion algebra; numerical linear algebra
Special Issues, Collections and Topics in MDPI journals
Dr. Dijana Mosić

E-Mail Website
Co-Guest Editor
Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia
Interests: generalized inverses; matrix equations; linear algebra; operator theory; functional analysis

Special Issue Information

Dear Colleagues,

We envision a collection of papers pertaining to advances in linear algebra and its applications. Linear algebra is known as the branch of mathematics concerning vector spaces and linear mappings between such spaces. However, linear algebra is the foundation to almost all areas of mathematics. Many ideas and methods of linear algebra have been generalized to abstract algebra, functional analysis, topology, etc. Systems of linear equations with several unknowns are naturally represented using the formalism of matrices and vectors. Functional analyses study the infinite-dimensional version of the theory of vector spaces. Matrix algebras over different areas, such as quaternion algebras, generate new features and applications. Recently, active research development has been observed in tensor algebra, which is a natural extension of matrix algebra. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Linear algebra is also used in most sciences and engineering areas, because it allows many natural phenomena to be modeled, and enables efficient computing with such models. This issue will present original studies in some leading areas of linear algebra and its applications.

Dr. Ivan I. Kyrchei
Dr. Zhuo-Heng He
Dr. Dijana Mosić
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. Axioms 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 algebra
  • matrix equation
  • quaternion matrix
  • generalized inverse
  • tensor
  • vector space
  • operator algebra

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Published Papers (3 papers)

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15 pages, 1379 KiB  
Article
RSVD for Three Quaternion Tensors with Applications in Color Video Watermark Processing
by Wen-Juan Chen and Shao-Wen Yu
Axioms 2023, 12(3), 232; https://doi.org/10.3390/axioms12030232 - 22 Feb 2023
Cited by 1 | Viewed by 1642
Abstract
In this paper, we study the restricted singular-value decomposition (RSVD) for three quaternion tensors under the Einstein product, and give higher-order RSVD over the quaternion algebra, which can achieve simultaneous singular value decomposition of three quaternion tensors. Moreover, we give the algorithm for [...] Read more.
In this paper, we study the restricted singular-value decomposition (RSVD) for three quaternion tensors under the Einstein product, and give higher-order RSVD over the quaternion algebra, which can achieve simultaneous singular value decomposition of three quaternion tensors. Moreover, we give the algorithm for computing the RSVD of for quaternion tensors. What is more, we present a new blind color video watermarking scheme based on the forth-order RSVD over the quaternion algebra, and our numerical example demonstrates the effectiveness of the framework. Full article
(This article belongs to the Special Issue Advances in Linear Algebra)
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16 pages, 349 KiB  
Article
A Fast Novel Recursive Algorithm for Computing the Inverse of a Generalized Vandermonde Matrix
by Ahmed Arafat and Moawwad El-Mikkawy
Axioms 2023, 12(1), 27; https://doi.org/10.3390/axioms12010027 - 26 Dec 2022
Cited by 3 | Viewed by 1948
Abstract
The main research object of this paper is to present a systematic computational procedure for computing the inverse of a generalized Vandermonde matrix. Short and rigorous proofs for the formulas of the determinant and the inverse of a generalized Vandermonde matrix are proposed. [...] Read more.
The main research object of this paper is to present a systematic computational procedure for computing the inverse of a generalized Vandermonde matrix. Short and rigorous proofs for the formulas of the determinant and the inverse of a generalized Vandermonde matrix are proposed. The computational cost of this method is O(n2). The proposed method can be used efficiently for hand calculation as well as for computer programming. Some examples are given for the sake of illustration. Furthermore, we present a simulation study to compare the time spent to calculate the inverse using the proposed algorithm and the inverse function in Maple. Full article
(This article belongs to the Special Issue Advances in Linear Algebra)
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28 pages, 339 KiB  
Article
Some Properties of the Solution to a System of Quaternion Matrix Equations
by Shao-Wen Yu, Xiao-Na Zhang, Wei-Lu Qin and Zhuo-Heng He
Axioms 2022, 11(12), 710; https://doi.org/10.3390/axioms11120710 - 8 Dec 2022
Viewed by 1221
Abstract
This paper investigates the properties of the ϕ-skew-Hermitian solution to the system of quaternion matrix equations involving ϕ-skew-Hermicity with four unknowns [...] Read more.
This paper investigates the properties of the ϕ-skew-Hermitian solution to the system of quaternion matrix equations involving ϕ-skew-Hermicity with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4. We present the general ϕ-skew-Hermitian solution to this system. Moreover, we derive the β(ϕ)-signature bounds of the ϕ-skew-Hermitian solution X1 in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have β(ϕ)-positive semidefinite, β(ϕ)-positive definite, β(ϕ)-negative semidefinite and β(ϕ)-negative definite solutions. Full article
(This article belongs to the Special Issue Advances in Linear Algebra)
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