Schur Complements And Special Matrix Product Matrix Inequalities

In the paper,we establish some matrix inequalities in howner partial ordering by means of Schur complement and matrix special product.The results mainly consist of two parts in the paper.In the first part,we establish some matrix inequalities by means of Schur complement and block matrix.We first prove the matrix equalities which obtain some matrix inequalities and determinantal inequalities.The Schur complement,which has been used by many other authors,serves as the main tool.Then,we show some new results about contraction.Finally, we present the matrix inequality involving Schur complement and Hadamard product.In the second part,we first obtain some matrix inequality involving matrix Hadamard product and Khatri-Rao product.The observation about an important connection between the Hadamard(Khatri-Rao) product and the Kronecker(Tracy-Singh) product Plays a key role in many matrix problems on the Hadamard(Khatri-Rao) product.And then,we simply extend the matrix inequalities in three ways that are proved by Fuzhen Zhang(F.Zhang, 2000),and obtain some existing and new matrix inequalities including famous inequalities such as Haynsworth inequality and Amemiya inequality by selecting special numbers and matrices in the new matrix inequalities.