Inequalities About Hermitian Matrix And The Rank Of A Class Of2×2Block Matrix

As a basic tool of mathematics, matrix has a wealth of research and a wide range of applications in probability and statistics, optimizing field, numerical analysis and other subjects. In the study of matrix theory, inequalities on the Hermitian matrix and inequalities about the rank of a class of2x2block matrix are very important.After learning matrix analysis, matrix inequality, topics of matrix analysis and other related courses, on the basis of absorbing and referencing to the previous results of many experts and scholars, this paper mainly analyzes the properties of Hermitian matrix eigenvalues, singular values and the rank of a class of2x2block matrix, and then discusses the inequalities on Hermitian matrix eigenvalues and singular values and the inequalities about the rank of a class of2x2block matrix. This paper is divided into four chapters, the main contents of each chapter are as follows:In the first Chapter, some of the mathematical symbols, definitions and lemmas which will be used in this paper are introduced. Firstly, the definitions about Hermitian matrix, positive definite matrix, positive semidefinite matrix, unitary matrix, similarity of matrix, the eigenvalue of the matrix, the singular value of the matrix, generalized inverse of the matrix are given; secondly, a number of important and related theorems are introduced.In the second Chapter, the relationships between Hermitian matrix eigenvalues and matrix transformation are studied, and then some inequalities on Hermitian matrix eigenvalues are gotten.In the third Chapter, firstly, properties of a class of positive semidefinite matrix are researched:secondly, this chapter gives some results about simultaneously diago-nalizable by congruence of two positive definite matrices and positive semidefinite matrices which satisfy certain conditions; finally, inequalities about singular values of positive definite matrix and positive semidefinite matrix are given.In the fourth Chapter, by using the relevant knowledge about Schur complement and generalized inverse, this chapter mainly gives inequalities about the rank of a class of2x2block matrix and gives an equivalent condition which is about the invariance of the perturbation of the rank.