一个特殊乘积的逆的矩阵不等式的加强

Strengthening of a matrix inequality on inverse of particular product

将两个正定矩阵的Khatri-Rao乘积的矩阵不等式(A*B)-1≤A-1*B-1推广为(A*B)-1≤(A-1(α)-1*B(α))-1 (A(α′)*B-1(α′)-1)-1≤(A-1(α)*B(α)-1) (A(α′)-1*B-1(α′))≤A-1*B-1,其中A(α)是A的顺序主子矩阵,而A(α′)是A(α)的余子矩阵.同时还给出了其等式成立的充分必要条件.

It is showed that the matrix inequality on inverse of Khatri\|Rao product of two positive definite matrices (A*B)\+\{-1\}≤A\+\{-1\}*B\+\{-1\} can be extended further to (AB) -1 ≤(A -1 (α) -1 *B(α))~-1 (A(α′)*B -1 (α′) -1 ) -1 ≤ (A -1 (α)*B(α) -1 )(A(α′) -1 *B -1 (α′))≤A -1 *B -1 , where A(α) is the leading principal submatrix of A ,and A(α′) is the complementary submatrix of A(α) .Furthermore,the necessary and sufficient condition that equalities hold is given.