四元数矩阵方程AXB=C通解中的复矩阵分量极秩

Extremal ranks of complex components in general solutions of the matric equation AXB=C over quaternion field

借助四元数矩阵的复表示方式Φ(·),将四元数体上的线性矩阵方程AXB=C转换为复数域上的等价复矩阵方程Φ(A)XΦ(B)=Φ(C).同时,利用该复矩阵方程的通解和分块矩阵的极秩性质,求出原四元数矩阵方程通解中复矩阵分量集{X0}和{X1}的最大秩、最小秩公式.作为这些极秩公式的应用,推导出了该四元数矩阵方程通解中包含复矩阵解或全为复矩阵解的充要条件.

By using a complex representation of quaternion matrix Φ(·),the linear matrix equation AXB=C over the quaternion field is changed into the matrix equationΦ(A)XΦ(B)=Φ(C)over the complex field.Then according to general solutions of this complex matrix equation and numerous properties regarding extreme ranks of block matrix, formulas of extreme ranks of complex matrices {X0},{X1} are established.These complex matrices are complex components of general solutions X=X0+X1j of the quaternion matrix equation.As an application,we give necessary and sufficient conditions for following special cases:there exists at least a complex matrix X in general solutions of the matrix equation;and all general solutions of the matrix equation are complex ones.