Theory And Calculations Based On The Generalized Inverse A <sub> T, S </ Sub> ~ (2) Generalized Schur Complement

The theory of generalized inverses and its computation, the theory of the Schur complements developed in the 1920s. There are a lot of results about them. The generalized inverses of a matrix have wide applications in many areas, such as differential and integral equations, operator theory, control theory, optimal theory, statistics, Markov chains, and etc. And Schur complement theory also has many applications in many areas, such as matrix theory, statistics analysis, numerical computation, the solution of a linear equation, region decomposition, linear control, and etc. Now, both of them are important tools in science computation. In this paper, firstly, we introduce the research history of Schur complement and Pseudo-Schur complement, and some recent results, then we defined the generalized Schur complement by different generalized inverses. The main work are as follows.The first part is just the second chapter, we give the retrospection about the research history of Schur complement and Pseudo-Schur complement, and some recent results on the rank of partitioned matrices, the proof of determinant relationship, the quotient property, we also give a brief introduction to the application of Schur complement.The second part consists of three chapters. In the third chapter, firstly, we define the generalized Schur complement based on the generalized inverse A_T,S², and other common generalized inverses such as the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse and the group inverse, thus extend the Pseudo-Schur complement concept. Secondly, we study its properties, applications and computation.In the fourth chapter, we study the property of this generalized Schur complement defined by the block idempotent matrices, we also studied the sum and difference of the generalized schur complement defined in the block idempotent matrices, the zero eigenvalue and the eigenvector of the zero eigenvalue, obtained some results about the idempotence of the generalized Schur complements.In the fifth chapter, we introduced four special matrix products, they are the Kronecker product, the Tracy-Singh product, the Khatri-Rao product and the Hadamard product. Combining the special matrices product with the generalized inverses, established a relationship between Khatri-Rao product and block matrices, studied the Tracy-Singh product of the {1} inverses and the {2} inverses, the Tracy-Singh product of oblique projections, obtained many new results by combining these matrices products with the generalized Schur complement.