scholarly journals Vector least-squares solutions for coupled singular matrix equations

2007 ◽  
Vol 206 (2) ◽  
pp. 1051-1069 ◽  
Author(s):  
Adem Kılıçman ◽  
Zeyad Abdel Aziz Al Zhour
Author(s):  
R. Penrose

This paper describes a generalization of the inverse of a non-singular matrix, as the unique solution of a certain set of equations. This generalized inverse exists for any (possibly rectangular) matrix whatsoever with complex elements. It is used here for solving linear matrix equations, and among other applications for finding an expression for the principal idempotent elements of a matrix. Also a new type of spectral decomposition is given.


Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.


Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2503-2520 ◽  
Author(s):  
Masoud Hajarian

The periodic matrix equations are strongly related to analysis of periodic control systems for various engineering and mechanical problems. In this work, a matrix form of the conjugate gradient for least squares (MCGLS) method is constructed for obtaining the least squares solutions of the general discrete-time periodic matrix equations ?t,j=1 (Ai,jXi,jBi,j + Ci,jXi+1,jDi,j)=Mi, i=1,2,.... It is shown that the MCGLS method converges smoothly in a finite number of steps in the absence of round-off errors. Finally two numerical examples show that the MCGLS method is efficient.


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