Abstract: Based on the hybrid steepest descent method，an iterative method for the symmetric solution and optimal approximation solution of matrix equations [(AXB=E,CXD=F)] with weighted norm is derived.For any initial matrix，a symmetric solution can be obtained within finite iteration steps in the absence of round-off errors.In addition，the related optimal approximation problem to a given matrix over the solution set is solved.

Key words: the hybrid steepest descent method, matrix equations, symmetric solution, weighted Frobenius norm, optimal approximation

摘要： 应用复合最速下降法，给出了求解矩阵方程组[(AXB=E,CXD=F)]加权范数下对称解及最佳逼近问题的迭代解法。对任意给定的初始矩阵，该迭代算法能够在有限步迭代计算之后得到矩阵方程组的对称解，并且在上述解集合中也可给出指定矩阵的最佳逼近矩阵。

关键词: 复合最速下降法, 矩阵方程组, 对称解, 加权Frobenius范数, 最佳逼近