四元数矩阵方程AXB+CXD=E的广义延拓解

本文在四元数体上讨论矩阵方程AXB+CXD=E的广义行（列）共轭延拓解问题.利用四元数矩阵的复与实分解，以及广义共轭延拓矩阵的结构特点，借助矩阵Kronecker积，把约束四元数矩阵方程转化为实数域上无约束方程，从而得到该方程具有广义行（列）共轭延拓解的充要条件及其通解表达式.最后通过数值算例说明所给算法的可行性.

THE CONJUGATED EXTENDED MATRIX SOLUTIONS OF THE QUATERNION EQUATION AXB + CXD=E

This paper is aimed at discussing the column and row conjugated extended matrix solutions of quaternion equation AXB + CXD = E. By using the complex and real decomposition of a quaternion matrix, the Kronecker product of matrices and the specific structure of a conjugated extended matrix, the quaternion equation with constraints can be converted to an unconstrained equation. Then the necessary and sufficient condition for the existence of the quaternion matrix equation AXB + CXD = E with column and row conjugated extended matrix and their general solution expression are obtained. Finally, the feasibility of the proposed algorithm will be illustrated through the numerical example.