Singular Value Inequalities On Some Matrix Classes

We focus on log-majorization relation on the singular values of partitioned positive semi definite matrices,the singular values of sums of matrices,the products of normal matrices and the rank of products of arbitrary matrices.The main results of this paper are divided into three parts.1.For partitioned positive semi definite matrix H=(?)?Mp+q with A E Mp and C E Mq(p?q).We prove that there exists a contractive matrix W?Mp×q,such that s(?)(?)log{si(I+?(W)si(A?C)}(p+q)(i=1),with ?(W):=?(?),which generalized classical Fischer's inequality:det(H)?det(A)det(C)=det(A ? C).Here“(?)log”stands for log-majorization.Meanwhile,we also give some singular value inequalities between partitioned positive semidefinite matrix H and diagonal matrix A ? C.In addition,some related and new inequalities are also obtained.2.A new strengthened form of singular value trigonometric inequality for the sums of matrices is given through the Jordan decomposition of the matrices,which is used to unify the two known singular value inequalities due to Bhatia,Kittaneh and Hirzallah,Kittaneh.Then,we discuss the relationship on the existing singular value inequalities.3.The product of normal matrices A and B:the singular value relationship between AB and BA is characterized.Furthermore,for any complex matrices A and B,we consider the maximum clearance between rank(AB)and rank(BA).