Schur Complements and Matrix Inequalities of Hadamard Products
Linear and Multilinear Algebra
Schur complement, Hadamard product, Principal submatrix
Let A and B be n-square positive definite matrices. Denote the Hadamard product of A and B by A o B. The main results of the paper are:
1. For any matrices C and D of size m×n
(C ο D)(A ο B)-1 (C ο D)* ≤ (CA-1 C*) ο (DB-1D*)
2. Let A/α be the Schur complement of A(α) in A. Then
(A ο B)/α ≥ A/α ο B/α.
Some other matrix inequalities of Schur complements and Hadamard products of positive definite matrices are also presented.
Wang, Bo-Ying and Zhang, Fuzhen, "Schur Complements and Matrix Inequalities of Hadamard Products" (1997). Mathematics Faculty Articles. 106.