Schur Complements and Matrix Inequalities of Hadamard Products

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*) and 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