复半正定矩阵的奇异值不等式

Some inequalities for singular values of complex positive semidefinite matrices

用Mn表示所有复矩阵组成的集合.对于A∈Mn,σ(A)=(σ1(A),…,σn(A)),其中σ1(A)≥…≥σn(A)是矩阵A的奇异值.本文给出证明:对于任意实数α,A,B∈Mn为半正定矩阵,优化不等式σ(A-|α|B) wlogσ(A+αB)成立,改进和推广了文[5]的结果.

Let M_n be the space of n×n complex matrices. For A∈M_n,let σ(A)=(σ_1(A),...,σ_n(A)),where σ_1(A)≥...≥σ_n(A) are the singular values of A.We prove that if A,B∈M_n are positive semidefinite,then σ(A-|α|B)__(wlog)σ(A+αB) hold for any real number α.This sharpens some results due to \.