徐玮玮

(南京信息工程大学数学与统计学院，南京 210044)

任意矩阵特征值的秩1修正扰动界

徐玮玮*

(南京信息工程大学数学与统计学院，南京 210044)

设A是一个n阶的任意复矩阵且E是A的Hermite秩1扰动，即E=xx′,其中x是n维的复列向量，x′是x的共轭转置向量.则A+E为矩阵A的Hermite秩1修正矩阵.基于矩阵分析理论中Hermite矩阵特征值分布的性质，研究得到了矩阵A特征值的任意Hermite秩1修正扰动的上下界限，即给出了矩阵A+E特征值的上下界限：

且

λmin(-SH(A)τ)≤S(λi(A+xx′))≤ λmax(-SH(A)τ)(1≤i≤n),

其中

gapi=λi-1(A)-λi(A),i=2,…,n,

特征值; 上下界; 秩1修正

and

λmin(-SH(A)τ)≤S(λi(A+xx′))≤ λmax(-SH(A)τ)(1≤i≤n),

where

Keywords：eigenvalue;two-sidebounds;rank-oneupdate

1　准备知识

xi:j≡(vi,…,vj)*x(i≤j).

令

且

(i=1,…,n-1),

其中 gapi为特征值和它的右邻特征值之间的距离, 即gapi=λi-1(A)-λi(A)(i=2,…,n).

下面将介绍一些有用的引理.

且

λi(A)+li(x)≤ λi(A+xx*)≤

min{λi(A)+ui(x), λi-1(A)}(2≤i≤n-1).

2　特征值修正扰动界

本节将考虑任意矩阵特征值的Hermite秩1修正扰动界. 首先定义矩阵复特征值的顺序: 令λi(B)(i=1,…,n)为B的特征值, λi(B)⪯λj(B)代表R(λi(B))≤R(λj(B))(i

λi(H(A))+li(x)+δi≤R(λi(A+xx*))≤

(1)

λi(H(A))+li(x)+δi≤R(λi(A+xx*))≤

(2≤i≤n-1)

(2)

且

λmin(-SH(A)τ)≤S(λi(A+xx*))≤

λmax(-SH(A)τ)(1≤i≤n),

(3)

Δz=z-φ(z), ω1=Δz*(H(A)+xx*)z,

ω2=φ(z)*(H(A)+xx*)Δz, ω3=z*SH(A)z.

那么

λi(A+xx*)=z*(A+xx*)z=

z*(H(A)+xx*)z+z*SH(A)z=(φ(z)+Δz)*(H(A)+xx*)(φ(z)+Δz)+z*SH(A)z=

φ(z)*(H(A)+xx*)φ(z)+ω1+ω2+ω3=

λi(H(A)+xx*)+ω1+ω2+ω3.

(4)

注意到R(ω3)=0和

S(Δz*(H(A)+xx*)Δz)=0.

由式(4)有

R(λi(A+yx*))= λi(H(A)+xx*)+

R(ω1)+R(ω2)

(5)

和

S(λi(A+yx*))=S(ω3).

(6)

易知R(ω1)+R(ω2)=z*H(A)z-λi(H(A)+xx*)+z x x*z.

(7)

若A是Hermite的, 那么Xi=Yi.由式(7)可知R(ω1)+R(ω2)=0.否则, 结合式(7)和引理1可得λmin(H(A))-λi-1(H(A))-ui(x)≤R(ω1)+R(ω2)≤

因此,

α(λmin(H(A))-λi-1(H(A))-ui(x))≤

R(ω1)+R(ω2)≤α(λmax(H(A))-

(8)

λmin(-SH(A)τ)≤S(ω3)≤ λmax(-SH(A)τ).

(9)

由引理1和式(5)～(9)可得式(1)～(3).

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【中文责编：庄晓琼英文责编：肖菁】

Eigenvalue Variations for Rank-one Update of Arbitrary Matrices

Xu Weiwei*

(School of Mathematics and Statistics, Nanjing University of information Science and Technology, Nanjing 210044,China)

Assume that matrix A is an arbitrary complex matrix of ordernand E is a Hermitian rank-one matrix, i.e., E=xx′, wherexis a complex column vector of ordernandx′ is the conjugate transpose vector ofx. So, A+E is called Hermitian rank-one update of matrix A. Based on the properties of matrix analysis theory in Hermitian matrix eigenvalue distribution, Hermitian rank-one perturbation bounds of an arbitrary matrix is given and two-side bounds for eigenvalues of A+E are presented as follows:

2014-07-11《华南师范大学学报(自然科学版)》网址：http://journal.scnu.edu.cn/n

江苏省自然科学青年基金项目(BK2013098);江苏省高校自然科学基金项目(13KJB110020)

徐玮玮，讲师，Email:wwx19840904@sina.com.

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1000-5463(2015)02-0158-03