四元数正定矩阵的等价表示

Equivalent Expressions of Quaternion Definite Matrices

四元数矩阵的理论，由于其在理论上和应用上的重要意义而引起人们的广泛关注，并取得了一系列的重要成果。而四元数正定（半正定）自共轭矩阵的理论无疑是这一理论的重要内容之一。作为四元数正定自共轭矩阵的推广，引入了正定四元数矩阵的概念，并利用四元数矩阵的复表示给出它的几个等价表示，从而将四元数正定矩阵的判定转化为复正定矩阵的判定或正定Ｈｅｒｍｉｔｅ矩阵的判定。因此，不仅这里所采用的研究方法是新颖的，而且所得结果的现形式也是新颖的。

The theory of Quaternion matrix aroused people′s broad concern because of its important significance in theory and application. A series important fruits are obtained. The theory of quaternion positive definite selfconjugate matrix is doubtlessly the important contents of this theory. As extence of quaternion positive definite selfconjugate matrix concept of quaternion positive definte matrix is introduced, and its several equivalent expressions are given. By using complex expression of quaternion matrix quaternion positive definite matrix was transformed into complex positive definite matrix. Therefore the method new is introduced here is quite original, the results obtained are quite new as well.