The Estimation Of Eigenvalue Or Trace Of The General Combination Of Hermite Matrix

As one of the most basic tools of mathematics, Matrix has a wealth of research contents and a long history of development.It has a wide range of application in optimizing field,probability and statistics, numerical analysis and mathematical physics, computational mathematics, biology, control theory, image processing, engineering technology, economics and Social Sciences. Recently,due to the development of mathematic and promotion of practices, the study of(semi-) positive definite matrix have been become an important subject in the matrix research fields. At present,(semi-) positive definite Hermite matrix has a wide range of application in control theory, optimization theory, matrix calculation,too. So, the study of(semi-) positive definite Hermite matrix is a active research topic and the study of its trace has a very important significance.After learning matrix analysis and other related courses, on the basis of absorbing and referencing to the previous research achievements of many experts and scholars, this paper mainly analyzes the properties of eigenvalue of Hermitian matrix and trace of(semi-) positive definite hermite matrix,and then it mainly discusses a series of inequality of the eigenvalue of Hermite matrix and the trace of(semi-) positive definite Hermite matrix. This paper is divided into four chapters, and the main contents of each chapter are as follows:In the first Chapter, we firstly introduce the research significance of the subject of the eigenvalue of matrix and trace of(semi-) positive definite matrix and overseas and domestic research actualities on this subject.At the same time,the paper gives a number of important and related theorems and concepts,and some mathematical symbol description;In the second Chapter, on the basis of referencing a number of inequalities of the eigenvalue of Hermite matrix of many experts and scholars,the paper takes from the particular to the general and classification discussion, and gets a series of inequalities of general combinations of eigenvalues of Hermite matrix;In the third Chapter, on the basis of previous research,we prove a series of inequalities of trace of(semi-) positive definite Hermite matrix.As extension,the paper proves a series of inequalities of trace of the real coefficient general combination of(semi-)positive definite Hermite matrix.On basis of this,as application,we are going to make some change on Holder inequality of trace of semi-positive definite Hermite matrix,then we apply the ideas to the general combination,and obtain some new inequalities.In the forth Chapter, we introduce the other method to study the eigenvalues and traces on the general combination of two Hermite matrice, and give the deference with the results in front.