A finite iterative method for solving a pair of linear matrix equations

2007 ◽  
Vol 189 (2) ◽  
pp. 1350-1358 ◽  
Author(s):  
Xingping Sheng ◽  
Guoliang Chen
Keyword(s):  
Iterative Method ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Linear Matrix Equations ◽  
Finite Iterative Method

scholarly journals Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yajun Xie ◽  
Changfeng Ma
Keyword(s):  
Iterative Method ◽  
Linear Matrix ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Initial Value ◽  
Least Frobenius Norm Solution ◽  
Centrally Symmetric ◽  
Solution Pair ◽  
Matrix Pair ◽  
Linear Matrix Equations

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair(X,Y)solutions of the generalized coupled Sylvester-conjugate matrix equationsA1X+B1Y=D1X¯E1+F1,A2Y+B2X=D2Y¯E2+F2. On the condition that the coupled matrix equations are consistent, we show that the solution pair(X*,Y*)can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient.

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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