关键词: 四元数 Ｌｙａｐｕｎｏｖ方程, 右特征值分解, 酉矩阵, 最佳逼近

Abstract: For the quaternion Lyapunov equation AX+XA*=C, under the A, C are normal matrix condition, by using the Frobenius norm unitary product invariance of quaternion matrix and the right eigenvalue decomposition of matrix, the necessary and sufficient conditions for the existence of the unitary structure solution of the equation and its general solution expression are obtained. At the same time, for a given unitary matrix M, the best approximation solution to M is obtained in the unitary structure solution set of the equation by using the trace inequality of the quaternion matrix and the block method of the matrix. Finally, the solution steps are given, and the correctness of the theoretical results and the feasibility of the given method are verified by numerical examples and complex representation operations of the quaternion matrix.