四元数矩阵方程AXB+CXD=E的M自共轭解

把实数域上的M对称矩阵的概念推广到四元数体上,形成M自共轭矩阵,然后在四元数体上讨论矩阵方程AXB+CXD=E的M自共轭解及其最佳逼近问题.利用四元数矩阵的实分解和复分解,以及M自共轭矩阵的特征结构,借助Kronecker积把约束四元数矩阵方程转化为实数域上的无约束方程,克服了四元数乘法非交换运算的困难,并得到该方程具有M自共轭解的充要条件及其通解表达式.同时在解集非空的条件下,运用矩阵的分块技术及矩阵的拉直算子,获得与预先给定的四元数矩阵有极小Frobenius范数的最佳逼近解.由于M自共轭矩阵是四元数自共轭矩阵的推广,因此所得结果拓展了该方程的结构解类型.

On M Self-Conjugate Solution of Quaternion Equation AXB+CXD=E

This paper aims at extending the concept of M symmetric matrix on real number field to the formation of M self-conjugate matrix on quaternion field and discussing M self-conjugate matrix solution of quaternion equation AXB+CXD=E and its optimal approximation. With the complex and real representations of a quaternion matrix, the Kronecker product of matrices and the specific structure of a M self-conjugate matrix, the quaternion equation with constraints can be converted to an unconstrained equation and to overcome the difficulty of non-commutative operation of quaternion multiplication. Then the necessary and sufficient conditions for the existence of the quaternion matrix equation AXB+CXD=E with M self-conjugate matrix and its general solution expression have been obtained. Meanwhile under the condition of the solution set of the M self-conjugate is not empty, by applying block matrix technology and matrix vec operator, and the expression of the optimal approximation solution to the given quaternion matrix is derived. Since M self-conjugate matrix is a generalization of self-conjugate quaternion matrix, the obtained results extend the type of structural solutions of this equation. Finally, we provide numerical algorithms and numerical examples to exemplify the results.