《矩阵理论—知识点详解》第四章 矩阵分解（4.4）Hermite矩阵特征值的变分特征

4 Hermite矩阵特征值的变分特征 定义:设A∈CM为 Hermite矩阵x∈C,称 H R(x)= x≠0 H 为A的 Rayleigh商

返回 4 Hermite矩阵特征值的变分特征 定义: 设 AC nn 为Hermite矩 阵, xC,称 ( ) = x  0 x x x Ax R x H H 为A的Rayleigh商

定理1( Rayleigh-Rtz): 设A∈CM为 ermita矩阵,则 (1)anr xsx Axsmxx (vrec (2)Amax =n1= max r(x)=max x Ax x≠0 H (3)amin =an=min R(x)=min x Ax x≠0 H

返回 定理1(Rayleigh-Ritz): 设 AC nn 为Hermite矩阵，则 (1) ( ) 1 H H H n n x x  x Ax   x x  x C R x x Ax H x x x H 0 1 (2) max 1 max ( ) max  =  =  = = R x x Ax H x x x n H 0 1 (3) min min ( ) min  =  =  = =

证:A为 Hermite矩阵→ A=UMU,A=dig(1,A2,…)Vx∈Cn x ax =x U AUx =(Ux)"A(Ux) J=Ux Ax =∑1|y i=1 →x"4x≥m.∑ly}=mny"y=mx"x →x" Ax<a∑|J2=1myy= H max min·x"xsx"Ax≤λmx:x"x

返回 证: A为Hermite矩阵 1 2 , ( , , ) H A U U diag =   =    n n  xC x Ax H H H =  x U Ux ( ) ( ) H =  Ux Ux 2 1 | | n H i i i x Ax y  = =  2 min 1 | | n H i i x Ax y  =   y Ux = min H =  y y min H =  x x 2 max 1 | | n H i i x Ax y  =   max H =  y y max H =  x x x x x Ax x x H H H  min     max 

定理2( Courant- Fischer):设A∈CN"为 Hermite 矩阵特征值为≤2≤…≤4k为给定的正 整数,1≤k≤n,则 mn max R()=nk n 01b n-k ∈ x≠0,x∈Cn x⊥a1,02.…On-k max min R()=nk ①1,02,…,Ok-1x≠0,x∈Cn x⊥a1 k-1

返回 k x C x x C R x n k n n n k        = − − ⊥    min max ( ) , , 0, 1, 2, 1, 2,   定理2(Courant -Fischer):设 为Hermite n n A C   矩阵, 特征值为1  2  n ，k为给定的正 整数，1  k  n，则 k x x x C R x k n k        = − − ⊥   max min ( ) 1, 2, 1 1, 2, 1 , 0, ,  

证:A为 Hermite矩阵 A=U AU, A=diag(n, n2,"An) xHAx (Ux)A(Ux R(x)=rHx (Ux)(Ux)

返回 证: A为Hermite矩阵 1 2 , ( , , ) H A U U diag =   =    n x x x Ax R x H H ( ) = ( ) ( ) ( ) ( ) H H Ux Ux Ux Ux  =