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中文摘要：
设矩阵X=（xij）∈R^n×n，如果xij=xn＋1-i，n＋1-j（i，j=1，2，…，n），则称X是中心对称矩阵．该文构造了一种迭代法求矩阵方程A1X1B1＋A2X2B2＋…＋AlXlBl=C的中心对称解组（其中[X1，X2，…，Xl]是实矩阵组）．当矩阵方程相容时，对任意初始的中心对称矩阵组[X1^（0），X2（0）．…，Xl^（0）]，在没有舍入误差的情况下，经过有限步迭代，得到它的一个中心对称解组，并且，通过选择一种特殊的中心对称矩阵组，得到它的最小范数中心对称解组．另外，给定中心对称矩阵组[^-X1，^-X2，…，^Xl]，通过求矩阵方程A1^～X1B1＋A2^～ X2B2＋…＋Al^～…XlBl=C（其中^～C=C～A1^-X1B1-A2^-X2B2-…-A1^-X1B1）的中心对称解组，得到它的最佳逼近中心对称解组．实例表明这种方法是有效的．
英文摘要：
A matrix X = （xij） ∈ R^n×n is said to be centrosymmetric if xij = Xn＋1-i,n＋1-j （i,j = 1,2,…,n）. In this paper, an iterative method is constructed for finding the centrosymmetrie solutions of matrix equation A1X1B1 ＋ A2X2B2 ＋ … ＋ AlXlBl = C, where [X1,X2…,Xl] is a real matrix group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial centrosymmetric matrix group[X1^（0）,X^2（0）, , … , Xl^（0）], a centrosymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm centrosymmetric solution group can be obtained by choosing a special kind of initial centrosylnmetric matrix group . In addition, the optimal approximation centrosymmetric solution group to a given centrosymmetric matrix group [^-X1, ^-X2,…,^-Xl] in Frobenius norm can be obtained by finding the least norm centrosymmetric solution group of new matrix equation A1^～X1B1 ＋ A2^～X2B2 ＋…＋ Al^～XlBl = ^～C,where ^～C = C - A1^～X1B1 - A2^～X2B2 Al^～XlBl. Given numerical examples show that the iterative method is efficient.
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