scholarly journals Algebraic Techniques for Least Squares Problems in Elliptic Complex Matrix Theory and Their Applications

2020 ◽  
Author(s):  
Hidayet Kösal ◽  
Müge Pekyaman
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Matrix Theory ◽  
Complex Matrix ◽  
Least Squares Problems ◽  
Real Numbers ◽  
Real Representation ◽  
Complex Matrices ◽  
Least Squares Solution ◽  
Elliptic Complex

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation AX=B by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known least squares solutions results from the literature.

scholarly journals Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

2021 ◽  
Vol 7 (4) ◽  
pp. 5029-5048
Author(s):  
Anli Wei ◽  
◽  
Ying Li ◽  
Wenxv Ding ◽  
Jianli Zhao ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Quaternion Matrix ◽  
Sylvester Matrix Equation ◽  
Real Representation ◽  
Minimal Norm ◽  
Generalized Sylvester Matrix Equation ◽  
Quaternion Matrix Equation ◽  
Special Solutions ◽  
Application Scope

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>

scholarly journals On the general strong fuzzy solutions of general fuzzy matrix equation involving the Core-EP inverse

2021 ◽  
Vol 7 (2) ◽  
pp. 3221-3238
Author(s):  
Hongjie Jiang ◽  
◽  
Xiaoji Liu ◽  
Caijing Jiang
Keyword(s):  
Least Squares ◽  
Unique Solution ◽  
Matrix Equation ◽  
Fuzzy Matrix ◽  
Least Squares Solution ◽  
The Core ◽  
Fuzzy Solutions

<abstract><p>The inconsistent or consistent general fuzzy matrix equation are studied in this paper. The aim of this paper is threefold. Firstly, general strong fuzzy matrix solutions of consistent general fuzzy matrix equation are derived, and an algorithm for obtaining general strong fuzzy solutions of general fuzzy matrix equation by Core-EP inverse is also established. Secondly, if inconsistent or consistent general fuzzy matrix equation satisfies $ X\in R(S^{k}) $, the unique solution or unique least squares solution of consistent or inconsistent general fuzzy matrix equation are given by Core-EP inverse. Thirdly, we present an algorithm for obtaining Core-EP inverse. Finally, we present some examples to illustrate the main results.</p></abstract>

scholarly journals Minimal properties of the Drazin-inverse solution of a matrix equation

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 383-395
Author(s):  
Marko Miladinovic ◽  
Sladjana Miljkovic ◽  
Predrag Stanimirovic
Keyword(s):  
Least Squares ◽  
Minimization Problem ◽  
Matrix Equation ◽  
Inverse Solution ◽  
Drazin Inverse ◽  
Minimal Norm ◽  
Least Squares Solution ◽  
The Matrix

We present the Drazin-inverse solution of the matrix equation AXB = G as a least-squares solution of a specified minimization problem. Some important properties of the Moore-Penrose inverse are extended on the Drazin inverse by exploring the minimal norm properties of the Drazin-inverse solution of the matrix equation AXB = G. The least squares properties of the Drazin-inverse solution lead to new representations of the Drazin inverse of a given matrix, which are justified by illustrative examples.

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