半正定矩阵广义Schur补的几个不等式

Some inequalities involving generalized Schur complements of positive semidefinite matrices

利用半正定矩阵的性质和矩阵Moore Penrose广义逆的特性,研究了半正定矩阵乘积及Hadamard积的广义Schur补的L wner偏序问题,得到了关于广义Schur补的若干不等式.对半正定矩阵A和B,给出了其Hadamard积广义Schur补与A/α B/α的关系,并对形如C AC(其中A半正定)的矩阵乘积,证明了(C AC)(β′)≥C (β′,α′)A/α·C(α′,β′)及(C AC)/α≤C /α·A(β′)·C/α.

Several inequalities in the lwner partial ordering involving product and Hadamard product are presented by using the properties of positive semidefinite matrices and their Moore-Penrose generalized inverse. For positive semidefinite matrices A and B, the relationship between (AB)/α and A/αB/α is studied. Furthermore, for matrix product of the from C~*AC where A is positive semidefinite, it′s proved that (C~*AC)(β′)≥C~*(β′,α′)A/α·C(α′,β′) and (C~*AC)/α≤C~*/α·A(β′)·C/α.