《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  =

{Ux:x∈C"且x≠0}={∈C":y≠0} n 1,02,…,On-k∈C maxR(x)=maX气 Ay x≠0,x∈Cm y≠0,y∈Cn H =,max∑4y1P yUa1…,Uo.,l=1

返回 1, 2, 1, 0, 0, , , max ( ) max n n n k n k H H x x C y y C x y U U y y R x y y      − −     ⊥ ⊥  = 1 , 2 1 1 , max | | H n k n i i y y i y U U y    − = = ⊥ =  { : 0} { : 0} n n Ux x C x y C y   =   且 n 1 ,2 ,  ,n−k C

≥max∑4|yP ⊥Ua1.…,Un-k V1=y yI -1=0 max ∑列12≥k y+1+…+n=1k y⊥Ua1…,UOnk

返回 1 , 1 2 1 0 2 1 1 , max | | H n k k n i i y y i y U U y y y y    − − = = = ⊥ = = =   2 2 2 1 1 , 2 | | | | | | 1 , max | | k k n n k n i i k y y y i k y U U y     + − + + + = = ⊥ =  

max R(x)≥k x≠0.x∈C x⊥01,02,,On-k ;=ln-i+1U=(1,u2,…,Ln) mn maX R(x)=nk 01b x≠0,x∈C x⊥1 2.'gOn-k

返回 k x x x C R x n k n      ⊥ −   max ( ) , 0, 1, 2,  ( , , , ) i = un−i+1 U = u1 u2  un k x x x C R x n k n n k        = − − ⊥   min max ( ) , 0, , 1, 2, 1, 2,  

定理3Wey)设A,B∈C为 Hermite矩阵则 Vk=1,2,…,n,有 k(A)+xn(B)≤k(4+B)≤Ak(A)+1(B) 证:x≠0,x∈C H Bx n(B)s~≤A1(B)

返回 定理3(Weyl):设 A,BC nn 为Hermite矩 阵,则 ( ) ( ) ( ) ( ) ( ) k A + n B  k A+ B  k A + 1 B k = 1,2,  ,n,有 证: n x  0, xC ( ) ( ) 1 B x x x Bx B H H n   

H LK(A+B) (A+ B)x min max O1…,0n-kx≠(0, H ⊥O1 k XAx x Bx min max x≠0 H H 01,…,0n-k ⊥1,…,0 H Ax ≥min max x≠0 .+n(B) n-k x⊥O1,,Cn2-k nK(A)+mn(B

返回 x x x A B x A B H H x x k n k n k ( ) ( ) min max , , , , 0, 1 1 + + = − − ⊥                 = + − − ⊥  x x x B x x x x A x H H H H x x n k n k     , , , , 0, 1 1 min max            + − − ⊥  min max ( ) , , , , 0, 1 1 B x x x A x H n H x x n k n k        (A) (B) = k + n