基于矩阵半张量积解四元数广义Sylvester矩阵方程组

Solving quaternion generalized Sylvester matrix equations based on semi-tensor product of matrices

该文利用矩阵半张量积求解四元数广义Sylvester矩阵方程组.首先将实矩阵半张量积运算推广到四元数矩阵,进而利用四元数矩阵半张量积提出四元数矩阵在向量算子下的一些新结论,利用这些结论将四元数矩阵方程组转化为四元数线性方程组,最后转化为实线性方程组,从而得到四元数广义Sylvester矩阵方程组有解的充要条件及通解表达式,并给出其极小范数解.最后通过数值算例说明该方法的有效性.

In this paper,the quaternion generalized Sylvester matrix equations are solved by using semi-tensor product of matrices.Firstly,semi-tensor product of real matrices is generalized to quaternion matrices,and then some new conclusions of quaternion matrix under vector operator are proposed by using semi-tensor product of quaternion matrices.Using these conclusions,quaternion matrix equations are transformed into quaternion linear equations,and finally into real linear equations.Consequently,the necessary and sufficient conditions for the compatibility and the expression of general solutions of equations are obtained,and the minimal norm solutions are given.Finally,numerical examples show the effectiveness of the method.