中文摘要：

讨论了矩阵方程（AX,XB）=（C,D）在线性流形上的次对称解及其最佳逼近.利用矩阵对构成新矩阵的奇异值分解导出了在线性流形上‖AX-C‖2＋‖XB-D‖2=min的最小二乘解及方程（AX,XB）=（C,D）存在次对称解的充分必要条件,并且给出了一般解的表达式及其它们的最佳逼近.

英文摘要：

This paper discussed the linear matrix equation（AX,XB）=（C,D） on linear manifold for the symmetric and its optimal approximation solution.The necessary and sufficient conditions for least-square solution of ‖AX-C‖2＋‖XB-D‖2=min on linear manifolds for the symmetric solution and sub-symmelric solution of the equation（AX,XB）=（C,D） are presented.Moreover,when solution sets of the above problems are nonempty,their general expressions are provided.The optimal approximation solutions in the solution set to a given matrix are obtained.