正定矩阵的Hadamard乘积的一个矩阵不等式的精细

A REFINEMENT OF MATRIX INEQUALITIES INVOLVING HADAMARD PRODUCTS OF POSITIVE DEFINITE MATRICES

周知的正定矩阵A和B的Hadamard乘积矩阵不等式 :(A B) -1 ≤A-1 B-1 被精细为(A B) -1 ≤diag((A-1 (α) -1 B(α) ) -1 ,(A(α′) B-1 (α′) -1 ) -1 ) ,≤diag(A-1 (α) B(α) -1 ,A(α′) -1 B-1 (α′) )≤A-1 B-1 ,这里A(α)是A的主子矩阵且α′是α的补序列 ;

The well-known matrix inequality (AB)^(-1)≤A^(-1)B^(-1) involving Hadamard product of positive definite matrices is refined to(AB)^(-1)≤diag((A^(-1)(α)^(-1)B(α))^(-1),(A(α′)B^(-1)(α′)^(-1))^(-1)),≤diag(A^(-1)(α)B(α)^(-1),A(α′)^(-1)B^(-1)(α′))≤A^(-1)B^(-1),where A(α) is the leading principal submatrix and α′ is the complementary sequence of α. The necessary and sufficient condition of these inequalities becomes equalities are presented.