四元数正定(半正定) 矩阵及四元数矩阵方程 AX= B的反问题

Quaternion Positive Definite(Half-Defininte)Matrix and Invese Problem of Quaternion Matrix Equation AX=B

实线性方程组 Ａ Ｘ＝ Ｂ 的反问题，由于它在控制理论中的重要应用而引起 人们的广泛关注，并取得了一系列重要成果。给出 ｎ 阶四元 数矩阵为正定（ 半 正定） 的充 要条件，研究 了四元数矩 阵方程 Ａ Ｘ＝ Ｂ 的反问题，得到 Ａ Ｘ＝ Ｂ 的反问题具有正定（ 半正定） 矩阵解、正定（ 半正定） 自共轭矩阵解的充要条件。另外，还给出了 Ａ Ｘ＝ Ｂ 的反问题的正定（ 半正定） 矩阵解与正定（ 半正定） 自共轭矩阵 解的一般形式。由于实数域和复数域是四元数体的子域。

The invese problem of the system of real liner equations AX=B aroused people′s broad concem because of its important application in the theory of control. And a series important results are obtained. In this paper, a necessary and sufficient condition is given for an n th order quaternion matrix to be positive definite(half-definite). The invese problem on quaternion matrix equation AX=B is discussed, a necessary and sufficient conditions is found for the invese problem of positive definite(half-definite) quaternion matrix and positive definite(half-definite) semiconjugate matrix. In addition, the general form of solutions of the quaternion matrix equation AX=B to have a solution of positive definite(half-definite) and positive definite(half-definite) semicojugate matries for the inverse problem of the quaternion matrix equation is also given. Since real number field and complex number field are sub-fields of the quaternion skew-field, therefore the results of some paper are special cases of results in this paper.