矩阵方程的解(Ⅲ)

利用熟悉的矩阵的秩研究了含两个未知矩阵和的矩阵方程的解的存在性,得到了通解结构,即:(X,Y)=ε +K1ε1+K2ε2+…+Krεr,其中ε1,ε2,…,εr为解空间S={(X,Y)｜AXB+CYD=0}的一个基,ε 为矩阵方程AXB+CYD=E的一个特解,K1,K2,…,Kr为任意常数,进一步讨论了矩阵方程AXB+CYD=E的解法.

Solution of the matrix equation AXB+CYD=E

*"By using the rank of matrix, the matrix equation with two unknown matrix X,Y is considered, the existence of solution is studied, and the structure of the general solution for this equation is obtained, which is as following: (X,Y)=ε+k1ε1+k2ε2+...+krε1rwhere ε1,ε2,...,εr are a basis for the solution spaceS={(X,Y)|AXB+CYD=0},ε is a particular solution of the matrix equation AXB+CYD=E, k1,k2,...,kr are arbitrary constants. Further, the method of solving the matrix equation AXB+CYD=E is discussed.