中文摘要：

给出了广义Sylvester矩阵方程AX—XF=BY当F为任意矩阵时的一种完全的解析通解．该通解由矩阵对（A，B）构成的能控性矩阵，一个对称算子矩阵和矩阵对（Z，F）构成的能观性矩阵组成，这里z是一个任意的参数矩阵，用来表征该方程的解的自由度．利用著名的Levverrier算法，该解析解的一个等价形式被给出．给出的结果是参考文献[13]的推广，在[13]中F被假设为友矩阵．

英文摘要：

A complete, general and explicit solution to the generalized Sylvester matrix equation AX -XF = BY, with F being an arbitrary square matrix, is investigated. The proposed solution is in an extremely neat form represented by a controllability matrix of the matrix pair （A ,B）, a symmetric operator and an observability matrix of the matrix pair （Z,F）, where Z is an arbitrary matrix used to denote the degree of freedom in the solution. Furthermore, based on the Faddeev - Leverrier algorithm, an equiva- lent form of the proposed solution is established. At the same time, an equivalent form of the solutions proposed in [ 13 ] is also induced. These results provide great convenience to the analysis and design problems in control systems. The results proposed in this note is a further discussion of the results proposed in [ 13 ].