矩阵方程AX－XB＝C的显式解──纪念导师郭仲衡教授

The Expliclt solution of the Matrix Equation AX-XB=C──To the memory of Prof

现有关于矩阵方程ＡＸ－ＸＢ＝Ｃ的显式解的几乎所有结论都是在Ａ与Ｂ无公共特征值的条件下获得的。本文利用特征投影给出了方程在Ａ与Ｂ均对称或反对称时一般解的显式形式。我们所得到的结果不仅适用于任何特征值重数情形，而且可以用来讨论该方程的一般情形。

Almost all of the existing results on the explicit solutions of the matriz equation AX-XB=C are obtained under the condition that A and B have noeigenvalues in common. For both symmetric or shewsymmetric matrices A and B,we shall give out the explicit general solutions of this equation by using thenotions of eigenprojections.The results we obtained.are applicable not only to anycases of eigenvalues regardless of their multiplicities,but also to the discussionof the general case of this cquation.