关于正定厄米特矩阵的几个不等式

Some Inequalities for Positive Definite Hermitian Matrices

本文推广了文献[1]、[3]给出的不等式,得到以下结果:(1)设Ai(i=1,2,…,k)都是n阶正定或半正定厄米特矩阵,p 1n,则|A1+…+Ak|p |A1|+…+|Ak|p;(2)设Ai,Bi,…,Ci(i=1,2,…,k)都是n阶正定或半正定厄米特矩阵,α,β…,r都是正实数,且α+β+…+r 1Ai｜α·｜Ai｜α·｜Bi｜β…｜Ci｜r ｜∑kn,则∑ki=1i=1Bi｜β…｜∑kCi｜r.

This paper generalizes some inequalities given in and ,and gets the following results:(1)If A1,A2,...,Ak are positive definite or positive semidefinite Hermitian matrices of order n,and p1n,|A1+...+Ak|p|A1|+...+|Ak|p;(2)If all Ai,Bi,...,Ci(i=1,...,k)are positive definite or positive semidefinite Hermitian matrices of order n,α>0,β>0,...,r>0,and α+β+...+r1n,then ∑ki=1|Ai|α*|Bi|β...|Ci|r|∑ki=1Ai|α*|∑ki=1Bi|β...|∑ki=1Ci|r.