Application Of Positive Semidefiniteness Of Block Matrix And Research Of Tripotent Matrix

This article was finished after learning some related courses of matrix, including matrix analysis, matrix inequality, matrix theory and some significant conclusions. It is composed of four chapters. The main contents of this paper are as follows:In Chapter1, we introduce some definitions, involving Hermitian matrices, positive semidefinite matrices, positive definite matrices, Hadmard products, idempotent matrices, unitary matrices, contractive matrices and similarity of matrix. Then we present a theorem which has played an important role in the second chapter.In Chapter2, we give the Cauchy-Schwarz inequality, the schur product theorem and the singular value decomposition theorem. These theorems act as lemmas for the further applications. Then some classical matrix inequalities are proved by using positive semidefiniteness of block matrices and the fundamental theorem in the first chapter.In Chapter3, we study the linear combinations of tripotent matrices by using linear combinations of idempotent Hermite matrices. Suitable conditions for the values of two coefficients are found.In Chapter4, Thompson theorem and the unitary decomposition theorem are introduced, some inequalities of trace are proved.