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4 Hermite矩阵特征值的变分特征
:定义 称矩阵为设,,H e r m i t e CxCA nn
0)( x
xx
AxxxR
H
H
的为 A,R a y le ig h 商返回
:)R i t z-R a y l e i g h(1定理矩阵，则为设 H e r m i t ennCA
)()1( 1 nHHHn CxxxAxxxx
AxxxR H
xxx H 10
1m a x m a x)(m a x)2(


AxxxR H
xxx
n H
10
m i n m i n)(m i n)3(


返回
:证 矩阵为 H e r m i t eA
12,(,,)H nA U U d i a g
nCx
Axx H HHx U U x ( ) ( )HU x U x
2
1
||
n
H
ii
i
x Ax y?

2
min
1
||
n
H
i
i
x Ax y?

y Ux?
mi n Hyy m i n Hxx
2
ma x
1
||
n
H
i
i
x Ax y?
m a x Hyy max Hxx
xxAxxxx HHH m a xm i n
返回
k
x
CxxC
xR
kn
nn
kn



)(m a xm i n
,
,0,
,2,1
,2,1
:)F i s c h e r-C o u r a n t(2定理 H e r m i t e为设 nnCA
为给定的正，征值为 kn21特矩阵,
，则整数,nk1
k
x
Cxx
xR
k
n
k



)(m i nm a x
1,2,1
1,2,1
,
,0,
返回
:证 矩阵为 H e r m i t eA
12,(,,)H nA U U d i a g
xx
AxxxR
H
H
)( ( ) ( )
( ) ( )
H
H
U x U x
U x U x

返回
1,2,1,
0,0,
,,
m a x ( ) m a x
nn
n k n k
H
Hx x C y y C
x y U U
yyRx
yy




1,
2
1 1
,
m ax | |
H
nk
n
ii
yy i
y U U
y



{,0 } {,0 }nnU x x C x y C y且
nkn C,,,21?
返回
1,
1 2 1 0
2
1
1
,
ma x | |
H
nk
k
n
ii
yy
i
y U U
y y y
y




2 2 2
1
1,
2
| | | | | | 1
,
ma x | |
k k n
nk
n
i i k
y y y ik
y U U
y




返回
k
x
Cxx
xR
kn
n



)(m a x
,
,0
,2,1?
),,,( 211 nini uuuUu
k
x
Cxx
xR
kn
n
kn



)(m axm i n
,
,0,
,2,1
,2,1
返回
:)W e y l(3定理 则矩阵为设,H e r m i t e,nnCBA
)()()()()( 1 BABABA kknk
有,,,2,1 nk
:证 nCxx,0
)()( 1 B
xx
BxxB
H
H
n
返回
xx
xBAx
BA
H
H
x
x
k
kn
kn
)(
m a xm i n)(
,,
,0,,
1
1




xx
Bxx
xx
Axx
H
H
H
H
x
x
kn
kn


,,
,0,,
1
1
m a xm i n

)(m a xm i n
,,
,0,,
1
1
B
xx
Axx
nH
H
x
x
kn
kn


)()( BA nk