广义Schur补的广义逆

Generalized Inverse of Generalized Schur Complement

对四分块矩阵A=A(︿) A(︿,︿′)A(︿′,︿) A(︿′)来说 ,如果 A和 A(︿)都是非奇异的 ,则A- 1 (︿′) =(A/︿) - 1 ,这里 A/ ︿=A(︿′) -A(︿′,︿) A(︿) - 1 A(︿,︿′)是 A(︿)在 A中的 Schur补 .王伯英教授指出上述等式 ,对半正定的 Hermitian矩阵而言 ,一般也是不能推广到 Moore-Penrose逆上去的 .在某些限制条件下 ,我们证明了广义逆的主子矩阵与广义 Schur补的关系是密切的 。

For four partitional matrix A=(A(α)A(α,α′)A(α′,α)A(α′))if A and A(α) are nonsingular, thenA -1(α′)=(A/α) -1,where A/α is Schur complement of A(α) in A Professor Wang BoYing point out that above equality d oes not generalize to the MoorePenrose inverse for positive semidefinite Herm itian Matrix. However, under some restiction, we prove that the relation between principal submatrix of generalized inverse and generalized Schur complement is close, therefore, classical result becomes special case.