复正定矩阵的若干性质

Some Characteristics of Complex Positive Definite Matrix

给出了复正定矩阵的一些等价特征及其有关的性质，如：复矩阵Ａ＝Ｈ＋ｉＳ（这里Ｈ和Ｓ均为Ｈｅｒｍｉｔｅ矩阵）是复正定的（复半正定的）当且仅当Ｈ是正定的（半正定的）；复矩阵Ａ是复正定的当且仅当存在可逆矩阵Ｐ使Ｐ＊ＡＰ＝Ｉｎ＋ｉＤｎ，其中Ｉｎ为ｎ阶单位矩阵，Ｄｎ为ｎ阶实对角矩阵。同时得到了复正定矩阵特征值的两个不等式。

Some equivalent characteristics of complex positive definite matrix are given, such as a complex matrix A=H+iS is complex positive definite (or complex positive semidefinite)if and only if H is positive definite(or positive semidefinite), where H and S are both Hermitian matrix; a complex matrix A is complex positive definite if and only if an invertible matrix P exists such that P* AP=In + iDn,where In is a unit matrix of n -order and Dn is a diagonal matrix and so on. Meanwhile,two inequalities of characteristic values with respect to complex matrix are obtained.