scholarly journals Least squares (P,Q)-orthogonal symmetric solutions of the matrix equation and its optimal approximation

2010 ◽  
Vol 20 ◽  
Author(s):  
Lin-lin Zhao ◽  
Guo-linag Chen ◽  
Qing-bin Liu
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Optimal Approximation ◽  
The Matrix

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

scholarly journals A class of iteration methods for the matrix equation A×B = C

2007 ◽  
Vol 75 (2) ◽  
pp. 289-298 ◽  
Author(s):  
Konghua Guo ◽  
Xiyan Hu ◽  
Lei Zhang
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Iteration Method ◽  
Optimal Approximation ◽  
Approximation Solution ◽  
Numerical Examples ◽  
The Matrix ◽  
Iteration Methods ◽  
Optimal Approximation Solution

An iteration method for the matrix equation A×B = C is constructed. By this iteration method, the least-norm solution for the matrix equation can be obtained when the matrix equation is consistent and the least-norm least-squares solutions can be obtained when the matrix equation is not consistent. The related optimal approximation solution is obtained by this iteration method. A preconditioned method for improving the iteration rate is put forward. Finally, some numerical examples are given.

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