Properties And Applications Of Schur Complement

Matrix Schur complement is one of the most important kens both in theory and applications, and it has wide applications in the study of Matrix Theory, Statistics and Analytics, Numerical Calculation, Linear equations, domain decomposition method, linear control etc.The corresponding applications of matrix Schur complement, generalized Schur complement, singular values of Schur complement of matrix products were discussed in this paper through some basics properties of matrix and mathematic methods.This paper is organized as follows: firstly, we give some necessary definitions and problems to be discussed in the introduction. In chapter 2, we clarify some properties of matrix Schur complement through using properties of block matrix, sub-matrix, idemfactor, etc. In chapter 5 and 6, we also show the algorithm and prove the veracity of the results.Secondly, we also discuss more Hermite matrix and properties of normal generalized Schur complement base on the results from Liu Jianzhou and Wang B.Y. We give a particular case of this algorithm as an example.At last, we obtain some inequalities, which can improve the eigenvalue inequations and the singularvalue inequations based on the singularvalues of Schur complement of matrix products that are research results of Marshall, A.W., Wang B.Y., and Liu Jianzhou. This paper also gives the applications of Linear equations and domain decomposition method.