Abstract: In this paper an iterative algorithm for solving the generalized Sylvester matrix equations is firstly  constructed by using the conjugate gradients and the matrix properties and the convergence of the algorithm is  proved.Furthermore,when the equation is compatible the algorithm is proved to converge to the minimal  norm solution to the problem.Then the algorithm is modified slightly to obtain the corresponding optimal approximation. Finally,numerical examples are given to verify the validity of the algorithm. 

Key words: Sylvester matrix equation, Conjugate gradient iterative method, Central symmetric class solution,  , Minimum norm solution, Optimal approximation