再论正定Hermite矩阵间的半正定关系

Semi-positive Definite Relation Between Two Positive Definite Hermitian Matrixes

文证明了两个交换ｎ阶正定Ｈｅｒｍｉｔｅ矩阵Ａ≥Ｂ的半正定Ｈｅｒｍｉｔｅ关系：１４（Ａ－Ｂ）２Ａ－１≤１２（Ａ＋Ｂ）－２ＡＢ（Ａ＋Ｂ）－１≤１４（Ａ－Ｂ）２Ｂ－１，１２（Ａ１２－Ｂ１２）２（Ａ－１Ｂ）１２≤（ＡＢ）１２－２ＡＢ（Ａ＋Ｂ）－１≤１２（Ａ１２－Ｂ１２）２（ＡＢ－１）

Following semipositive definite relation between two exchangeable positive definite Hermitian matrixes A≥B of ordern are proved: 14(A-B) 2A -1 ≤12(A+B)-2AB(A+B) -1 ≤14(A-B) 2B -1 , 12(A 12 -B 12 ) 2(A -1 B) 12 ≤(AB) 12 -2AB(A+B) -1 ≤12(A 12 -B 12 ) 2(AB -1 ) 12 , Where necessary and sufficient condition for holding egual sign is A=B.