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

作　　者：杨忠鹏[1] 

机构地区：[1]莆田学院数学系,福建莆田351100

出　　处：《高校应用数学学报（A辑）》2003年第4期473-479,共7页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：福建省教育厅科研基金项目(JB01206)

摘　　要：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.

关 键 词：矩阵不等式 HADAMARD乘积 Khatri—Rao乘积 正定矩阵 等式条件 矩阵分块 逆矩阵 

分 类 号：O151.21[理学—数学]