Some Majorization Inequalities For Positive Semidefinite Matrices

In the field of pure mathematics research,it is important to discuss the relationship between variables.In the past thirty years,the problem of matrix inequality has become a core research field in matrix theory.We study some problems on positive semidefinite block matrices,unitary invariant norm inequalities,the sums of matrices and the sums of corresponding absolute values of matrices.Our main results are as follows:1.We obtain some eigenvalue majorization inequalities for positive semidefinite block matrix.We apply the obtained inequalities and generalize the main theorems in some literatures so that they can be more widely used in other scientific fields.2.Using the block matrix technique,we study the Holder-type inequality between the sum of the matrix and the sum of its modulus and the continuous function inequality between the sum of the matrix and its modulus.3.We establish a weak log-majorization inequality on the singular values of the block PPT matrix.