Matrix Singular Value Eigenvalue Inequalities And Matrix Inequalities,

The matrix theory as a basic tool of mathematical subjects, are widely used in mathematics and other disciplines of science and technology fields (such as numerical analysis, optimization theory, differential equations, probability and statistics, systems engineering, etc.). But in the study of matrix theory, the inequalities for eigenvalues , singular values and matrices are important undoubtedly.This article is completed after my learning matrix theory and related programmes, absorbing in and drawing on research results of the well-known matrix theoryscholars. This paper is divided into three parts.In the first part,I will obtained a series of majorization inequalities and inequalities for singular values of ordinary and power products of formal matrices by means of the relations between eigenvalues and singular values and the properties of the formal matrices and compound matrices on the basis of the reference [6, Bo-Ying Wang and Ming-Peng Gong, 1993], and extend some conclusions of the reference [6, Bo-Ying Wang and Ming-Peng Gong,1993].In the second part I obtained some inequalities for eigenvalus and singular values of two positive semidefinite matrices products, and then give another way to prove some theories in the reference [6, Bo-Ying Wang and Ming-Peng Gong, 1993] by the properties of compound matrices and the relations between minimal eigenvalue and eigenvalue.In the last part of this paper ,first, I will obtained some inequalities for eigenvalues of Schur complements of matrices by means of Schur complements on the basis of the lemma 4.2.11 ;and then obtaind some matrices inequalities of Schur complements of the sum and Hadamard Products of matrices; Finally, I obtaind some matrix inequalities of Schur complements and principal submatrix of matrices by means of the relations between Schur complements and principal submatrix of matrices.