Styan不等式的推广

An Extension of Styan Inequality

设ｘ为ｎ×１单位向量，Ａ为ｎ×ｎ正交阵，Ｓｔｙａｎ［１］证明了这里λ１和λｎ为Ａ的最大和最小特征值。本文的目的是将上式的ｘ推广为ｎ×ｐ矩阵Ｘ，即证明了，对一切满足Ｘ′Ｘ＝Ｉｐ的矩阵ｘ和正定阵Ａ这里ｎ＞２ｐ，λ１≥…≥λｎ为Ａ的特征值，ｔｒ（Ａ）表示方阵Ａ的迹。

Let x be x×ι unit vector,and A an n×n positive definite matrix,styan [1] proved where λ1 and λn are the largest and the smallest eigenvalues of A.This paper extends this inequality and proves that where X is an n×p matrix with X'X=Ip and n>2p.λ1≥…≥λn are the eigenvalues of positive definite matrix A.