一类四元数矩阵方程组通解复分量的极秩

Extremal Ranks of Complex Matrices in General Solutions of a System of Quaternion Matrix Equation

【目的】在四元数体上研究矩阵方程组[AX XD]=[B E]的通解复分量极秩问题。【方法】利用四元数矩阵的复表示将原方程组转化为等价的复方程组,再通过复矩阵奇异值分解获得等价方程的通解表达式。【结果】根据分块矩阵秩的关系得到原方程组通解的复分量极秩计算公式,并在方程组无解时得到最小二乘复矩阵解的极秩公式。【结论】结果拓展了四元数体上方程组解的极秩理论并获得复分量表示通解的极秩计算方法。

[Purposes]It researches the problem on extremal ranks of complex component in general solution of a quaternion system[AX XD]=[B E].[Methods]By using the complex representation of a quaternion matrix and singular value decomposition of a complex matrix,the equations are transformed into equivalent complex matrix equations,and its expressions of the general solution and least square solution are given.[Findings]Maximal and minimal ranks of the complex component in general solution are obtained from rank identity of block matrix.[Conclusions]The results extend the extreme rank theory and its calculation method of the solutions of equations over the quaternion field.