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Axioms
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Advances in Linear Algebra with Applications, 2nd Edition
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Advances in Linear Algebra with Applications, 2nd Edition

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

Deadline for manuscript submissions: 30 June 2025 | Viewed by 1477

Special Issue Editor

Prof. Dr. Jorge Delgado Gracia

E-Mail Website
Guest Editor
Departamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
Interests: numerical linear algebra; computer-aided geometric design; approximation theory; numerical analysis
Special Issues, Collections and Topics in MDPI journals

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 this 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 applications of methods of numerical linear algebra in Science, Engineering, and Economics are welcome.

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

Prof. Dr. Jorge Delgado Gracia
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. 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 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 (4 papers)

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Research

18 pages, 266 KiB  
Article
The Reverse Order Law for the {1,3M,4N}—The Inverse of Two Matrix Products
by Yingying Qin, Baifeng Qiu and Zhiping Xiong
Axioms 2025, 14(5), 344; https://doi.org/10.3390/axioms14050344 - 30 Apr 2025
Viewed by 94
Abstract
By using the maximal and minimal ranks of some generalized Schur complement, the equivalent conditions for the reverse order law (AB){1,3M,4K}=B{1,3N,4K}A{1,3M,4N} are presented. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)
13 pages, 283 KiB  
Article
High Relative Accuracy for Corner Cutting Algorithms
by Jorge Ballarín, Jorge Delgado and Juan Manuel Peña
Axioms 2025, 14(4), 248; https://doi.org/10.3390/axioms14040248 - 26 Mar 2025
Viewed by 182
Abstract
Corner cutting algorithms are important in computer-aided geometric design and they are associated to stochastic non-singular totally positive matrices. Non-singular totally positive matrices admit a bidiagonal decomposition. For many important examples, this factorization can be obtained with high relative accuracy. From this factorization, [...] Read more.
Corner cutting algorithms are important in computer-aided geometric design and they are associated to stochastic non-singular totally positive matrices. Non-singular totally positive matrices admit a bidiagonal decomposition. For many important examples, this factorization can be obtained with high relative accuracy. From this factorization, a corner cutting algorithm can be obtained with high relative accuracy. Illustrative examples are included. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)
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9 pages, 384 KiB  
Article
Linear Algebraic Approach for Delayed Patternized Time-Series Forecasting Models
by Song-Kyoo Kim
Axioms 2025, 14(3), 224; https://doi.org/10.3390/axioms14030224 - 18 Mar 2025
Viewed by 247
Abstract
This paper introduces a linear algebraic approach for forecasting time-series trends, leveraging a theoretical model that transforms historical stock data into matrices to capture temporal dynamics and market patterns. By employing an analytical approach, the model predicts future market movements through delayed patternized [...] Read more.
This paper introduces a linear algebraic approach for forecasting time-series trends, leveraging a theoretical model that transforms historical stock data into matrices to capture temporal dynamics and market patterns. By employing an analytical approach, the model predicts future market movements through delayed patternized time-series machine learning training, achieving an impressive accuracy of 83.77% across 10,539 stock data samples. The mathematical proof underlying the framework, including the use of validation matrices and NXOR operations, ensures a structured evaluation of predictive accuracy. The binary trend-based simplification further reduces computational complexity, making the model scalable for large datasets. This study highlights the potential of linear algebra in enhancing predictive models and provides a foundation for future research to refine the framework, incorporate external variables, and explore alternative machine learning algorithms for improved robustness and applicability in financial markets. The primary advantages of employing linear algebra in this research lay in its ability to systematically structure high-dimensional financial data, enhance computational efficiency, and enable rigorous validation. The results indicate not only the efficacy in trend forecasting but also its potential applicability across various financial settings, making it a valuable tool for investors seeking data-driven insights into market trends. This research paves the way for future studies aimed at refining forecasting methodologies and enhancing financial decision-making processes. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)
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27 pages, 1810 KiB  
Article
Efficient Tensor Robust Principal Analysis via Right-Invertible Matrix-Based Tensor Products
by Zhang Huang, Jun Feng and Wei Li
Axioms 2025, 14(2), 99; https://doi.org/10.3390/axioms14020099 - 28 Jan 2025
Viewed by 593
Abstract
In this paper, we extend the definition of tensor products from using an invertible matrix to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. Based on this novel definition, we define the concepts of tensor rank and tensor nuclear [...] Read more.
In this paper, we extend the definition of tensor products from using an invertible matrix to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. Based on this novel definition, we define the concepts of tensor rank and tensor nuclear norm, ensuring consistency with their matrix counterparts, and derive a singular value thresholding (∗L,R SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is integral to our proposed tensor robust principal component analysis (∗LR TRPCA) algorithm. The computational complexity of the ∗LR TRPCA algorithm is O(k·(n1n2n3+p·min(n12n2,n1n22))) for k iterations. According to this complexity analysis, by using a right-invertible matrix that selects p rows from the n3 rows of the invertible matrix used in the tensor product with an invertible matrix, the computational load is approximately reduced to p/n3 of what it would be with an invertible matrix, highlighting the efficiency gain in terms of computational resources. We apply this efficient algorithm to grayscale video denoising and motion detection problems, where it demonstrates significant improvements in processing speed while maintaining comparable quality levels to existing methods, thereby providing a promising approach for handling multi-linear data and offering valuable insights for advanced data analysis tasks. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)
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