矩阵方程X-AXB^T=C的一般解

Solution of the Matrix Equation X-AXBT=C

在常微分方程组及其稳定性理论、控制论、线性统计等领域中会涉及到下列类型的线性矩阵方程Ｘ－ＡＸＢＴ＝Ｃ。当该矩阵方程满足相容性条件时，可利用矩阵的Ｋｒｏｎｅｃｋｅｒ积把它化为线性方程组（ＩＩ－ＡＢ）Ｘ＝Ｃ，再利用矩阵的Ｊｏｒｄａｎ分解式和分块矩阵的广义逆理论给出该矩阵方程的一般解。

The main purpose of this paper is to obtain solution linear matrix equation X-AXBT=C.This kind of equation often appears in the their stabilty properties, in the control theory in the linear statistical inference and so on. When this equation is consistent, we make use of Kronecker products to view the equation as a system of linear equations (II-AB)= and obtain its solution by means of Jordan decomposition and generalized inverses of partitioned matrices. 