中文摘要：

利用自共轭四元数矩阵的酉对角分解和正负惯性指数为工具,刻划了四元数体上离散型Lyapunov方程AXA＊-X=F的反问题有解的充要条件,并得到解A的具体表达式。同时,给出该方程在酉矩阵约束条件下的解和最小二乘解。推广了对称双侧正交Procrustes问题。

英文摘要：

In this paper, we have used diagonalization method and inertial exponent of a self-conjugate qua- ternion matrix, the necessary and sufficient conditions for the existence a solution of inverse problem of the discrete Lyapunov equation AXA＊-X=F over quaternion field have been given, and obtain the expres- sion of general solution A. Meanwhile, the least squares solution of the equation in quaternion unitary ma- trices set has been presented. We have generalized the Procrustes problem with bilateral orthogonal.