中文摘要：

给出一个迭代算法求解线性矩阵方程sum from i=1 to N（A_lX_lB_l=C）的对称解X1,X2,…,XN,利用这个迭代算法可以判断这个方程是否有对称解。当矩阵方程相容时,可以通过有限步迭代之后得到它的对称解;当选择特定的初始值时,迭代之后得到的是其极小范数对称解;此外,通过求新线性矩阵方程的极小范数对称解能够得到给定矩阵的最优逼近解。最后给出了一个数值例子来验证结论。

英文摘要：

A finite iterative algorithm was proposed to solve for the symmetric solutions（X1,X2,…,XN） of the matrix equations sum from i=1 to N（A_lX_lB_l=C）.If the matrix equation is consistent,the symmetric solutions could be obtained within finite iterative steps and its least-norm symmetric solution could be reached by choosing a special kind of initial iterative matrix.Furthermore,its optimal approximation solution to a given matrix can be derived by computing the least-norm symmetric solution of a new matrix equation.Finally,a numerical example was illustrated to verify the theoretical results.