scholarly journals The matrix iterative methods for solving a class of generalized coupled Sylvester-conjugate linear matrix equations

2015 ◽  
Vol 39 (16) ◽  
pp. 4895-4908 ◽  
Author(s):  
Ya-Jun Xie ◽  
Chang-Feng Ma
Keyword(s):  
Iterative Methods ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
Linear Matrix Equations

scholarly journals The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu
Keyword(s):  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
The Common ◽  
Necessary And Sufficient ◽  
Nonnegative Definite ◽  
Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

Convergence of ADGI methods for solving systems of linear matrix equations

2014 ◽  
Vol 31 (4) ◽  
pp. 681-690 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Iterative Methods ◽  
Design Methodology ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Hierarchical Identification ◽  
Content Type ◽  
Alternating Direction ◽  
Gradient Based ◽  
Engineering Physics ◽  
Linear Matrix Equations

Purpose – The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations. Design/methodology/approach – According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations. Findings – The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations. Originality/value – This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.

On the matrix equations U_iXV_j W_{i j} for 1 \leq i; j \leq k with i +j \leq k

2016 ◽  
Vol 31 ◽  
pp. 465-475
Author(s):  
Jacob Van der Woude
Keyword(s):  
Linear Matrix ◽  
Matrix Equations ◽  
Kronecker Products ◽  
Solvability Conditions ◽  
Common Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Triangular Decoupling ◽  
The Given ◽  
Linear Matrix Equations

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j 􏰁 W_{ij} for 1 \leq 􏰃 i,j \leq 􏰂 k with i\leq 􏰀 j \leq 􏰃 k, where the given matrices U_i,V_j,W_{ij} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability conditions, stated directly in terms of the given matrices and not using Kronecker products, are also presented. As an application, a version of the almost triangular decoupling problem is studied, and conditions for its solvability in transfer matrix and state space terms are presented.

scholarly journals The Solutions of Second-Order Linear Matrix Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Kefeng Li ◽  
Chao Zhang
Keyword(s):  
Time Scales ◽  
Characteristic Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Existence Of A Solution ◽  
Order Linear ◽  
The Matrix ◽  
Linear Matrix Equations

This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.

scholarly journals A generalized inverse for matrices

1955 ◽  
Vol 51 (3) ◽  
pp. 406-413 ◽  
Author(s):  
R. Penrose
Keyword(s):  
Unique Solution ◽  
Spectral Decomposition ◽  
Generalized Inverse ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Singular Matrix ◽  
Rectangular Matrix ◽  
New Type ◽  
Linear Matrix Equations

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.

scholarly journals The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 607-613 ◽  
Author(s):  
Xiang Wang ◽  
Dan Liao
Keyword(s):  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Convergence Factor ◽  
Gradient Based ◽  
Gradient Type ◽  
And Mathematics ◽  
Computers And Mathematics ◽  
Linear Matrix Equations

A hierarchical gradient based iterative algorithm of [L. Xie et al., Computers and Mathematics with Applications 58 (2009) 1441-1448] has been presented for finding the numerical solution for general linear matrix equations, and the convergent factor has been discussed by numerical experiments. However, they pointed out that how to choose a best convergence factor is still a project to be studied. In this paper, we discussed the optimal convergent factor for the gradient based iterative algorithm and obtained the optimal convergent factor. Moreover, the theoretical results of this paper can be extended to other methods of gradient-type based. Results of numerical experiments are consistent with the theoretical findings.

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