一四元数矩阵方程组的最佳逼近解

The Optimal Approximation Solutions to a System of Matrix Equations over the Quaternion Field

利用四元数矩阵的广义Frobenius范数和弱圈积,建立一个关于四元数矩阵的实函数并简洁表征其极小值.再用四元数矩阵的奇异值分解和广义Frobenius范数的性质,讨论四元数矩阵方程组[AX,XB]=[C,D]的最小二乘解,得到了解的具体表达式.最后在该方程组的解集合中导出了与给定矩阵的最佳逼近解的表达式.

By using generalized Frobenius norm and weak loop product of the quaternion matrices, a real-valued function for the quaternion matrices are established and its minimal value is simply expressed. Then using the singular value decomposition of the quaternion matrices and property of the generalized Frobenius norm, least-square solutions of the quaternion matrix equations [AX, XB]=[C, D] is discussed, and the expression of the solution is obtained. In addition, in the solution set of this matrix equations, the expression of the optimal approximation solution to a given matrix is provided.