On the gradient-based algorithm for solving the general coupled matrix equations

Recently, the gradient-based iterative algorithms have been widely exploited for finding the (least-squares) solutions of the different kinds of (coupled) linear matrix equations. Nevertheless, so far, the convergence of the propounded gradient-based algorithms has been studied for the case where the mentioned (coupled) linear matrix equations have a unique (least-squares) solution. In the present paper, we consider the consistent general coupled linear matrix equations which incorporate many of the recently investigated (coupled) linear matrix equations as their special instances. It is demonstrated that using a gradient-based iterative algorithm for solving the mentioned coupled linear matrix equations is equivalent to extending the well-known Richardson method for solving the normal equations corresponding to the original coupled linear matrix equations. In addition, we prove the semi-convergence of the Richardson method when the coefficient matrix of the associated normal equations is singular. Finally, some numerical experiments are presented to illustrate the validity of our theoretical results.