Inequalities of block matrices in the context of the schur complement

Schur complement is a key tool in many areas of matrix analysis and a rich source for matrix inequalities. The use of the Schur complement technique has an algorithmic structure in calculating matrix inverses and linear systems. In this study we examine equalities and inequalities involving the Schur complement of the large sized matrices. For this purpose we use positive definite and positive semi definite block matrices and examine the Schur complement of their products, such as Hadamard product, Kronecker product. Also in this thesis we analyze the block form of Hadamard and Kronecker products. We make an analogy between the inequalities involving Hadarmard, Kronecker products and block form of these products.