Cauchy不等式矩阵形式的推广

Generalizations of Cauchy inequality about matrix versions

设A为n×n正定Hermite阵,x为n维列向量,得到了Cauchy不等式的推广形式,进一步设A为n×n半正定Hermite阵,若x∈μ(A),推广形式仍成立.将向量x为n维列向量推广为X为n×p矩阵,且满足X*X=Ip,则有进一步推广.

Let A be an n×n positive definite Hermite matrix,x be a vector of dimension n, the generalized Cauchy inequality is obtained. And more, let A be an n×n semi-positive definite Hermite matrix and x ∈μ （A）, the inequality is established also. Thirdly, let X is n×p real matrix and is X^＊X =Ip, the generalized Cauchy inequality is obtained.