第十四周二课堂内容概要

14

14.1

( ) Æ Æ Æ

Æ Æ Æ ( ) Æ ( ) Æ ( )

Æ

14.1.1

( ). (V,hi)DZ V A ( )

A∗◦A=A◦A∗

( ). A∈Mn(C)∪Mn(R)A ( ) A∗·A=A·A∗

A A

A=diag(2i,2,1,···,1)

DZ

(~b1,···,~bn) (V,hi) AV

1≤i≤nA(~b)=λ~b,(λ

∈C) 1≤i≤nA(b

∗~ ~

i i i i

1≤i≤n

i)=λibi

λih~bi|~bii=hλi~bi|~bii=hA(~bi)|~bii=h~bi|A∗(~bi)i

λih~bi|~bii=hA(~bi)|~bii=h~bi|A∗(~bi)i

h~bi|A∗−λiE(~bi)i=h~bi|A∗(~bi)i−λih~bi|~bii=0

1≤j≤n,j⊕=i0=λjh~bj|~bii=hA(~bj)|~bii=h~bj|A∗(~bi)i

h~bj|A∗−λiE(~bi)i=h~bj|A∗(~bi)−λi~bii=h~bj|A∗(~bi)i−λih~bj|~bii=0

A∗−λiE(~bi)=~0DZ A∗(~bi)=λi~bi

(1)A A−λE(λ∈C)

(2) A λ∈C ~x∈VA(~x)=λ~x⇐⇒A∗(~x)=λ~x

DZ(λE)∗=λE (A−λE)∗=A∗−λE

A

kA(~x)k2=hA(~x)|A(~x)i=h~x|A∗A(~x)i=h~x|AA∗(~x)i=hA∗(~x)|A∗(~x)i=kA∗(~x)k2kA(~x)−λ~xk=kA∗(~x)−λ~xk

( Æ ). (1) (V,hi) AV

Æ ADZV

(2)A∈Mn(C) Æ A

()ÆÆÆ

ÆÆÆ()Æ()Æ()

Æ

14.1().(V,hi)DZVA()

∗

∗

A◦A=A◦A

14.2().A∈Mn(C)∪Mn(R)A()A∗·A=A·A∗

14.3.AA

A=diag(2i,2,1,···,1)

DZ

14.4.(~b1,···,~bn)(V,hi) AV

∗~ ~

1≤i≤nA(~bi)=λi~bi,(λi∈C)1≤i≤nA(bi)=λibi

1≤i≤n

λih~bi|~bii=hλi~bi|~bii=hA(~bi)|~bii=h~bi|A∗(~bi)i

λih~bi|~bii=hA(~bi)|~bii=h~bi|A∗(~bi)i

h~bi|A∗−λiE(~bi)i=h~bi|A∗(~bi)i−λih~bi|~bii=0

1≤j≤n,j⊕=i0=λjh~bj|~bii=hA(~bj)|~bii=h~bj|A∗(~bi)i

h~bj|A∗−λiE(~bi)i=h~bj|A∗(~bi)−λi~bii=h~bj|A∗(~bi)i−λih~bj|~bii=0

A∗−λiE(~bi)=~0DZA∗(~bi)=λi~bi

14.5. (1)AA−λE(λ∈C)

(2)Aλ∈C~x∈VA(~x)=λ~x⇐⇒A∗(~x)=λ~x

(1)DZ(λE)∗=λE (A−λE)∗=A∗−λE

(2)A

2 ∗ ∗ ∗ ∗ ∗ 2

kA(~x)k=hA(~x)|A(~x)i=h~x|AA(~x)i=h~x|AA(~x)i=hA(~x)|A(~x)i=kA(~x)k

kA(~x)−λ~xk=kA∗(~x)−λ~xk

14.6(Æ). (1)(V,hi)AV

ÆADZV

(2)A∈Mn(C)ÆA

153

AV (~b1,···,~bn) Æ diag(λ1,···,λn)

A(~bi)=λi~bi(1≤i≤n)

14.4 1≤i≤nA∗(~bi)=λi~bi 1≤i≤n

(AA∗−A∗A)(~bi)=~0

AA∗=A∗A

A (V,hi) λi∈Spec(A)

Wi={~x∈V|A(~x)=λi~x}

∗

...............................................................................................................................................................................................................................................................................................................................................................................................................................14.5WiDZA

W⊥ A DZA∗

i

i

~y∈W⊥ x~∈Wi

hA(y)|~xi=hy~|A∗(x~)i=0

∗∗

⊥

∗

(A)=A

WiDZ

A

i i

A A∗ W⊥ A W⊥ DZ

Wi=⊕V A W⊥i Æ V=Wi⊕W⊥i

( Æ ). (1)A∈L(V) (V,hi)A V DZ Æ

diag a1 b1

,···, ar br

,λ ,···,λ

−b1 a1

−br ar

2r+1 n

bi>0(1≤i≤r)A Æ DZ

√

ai±−1bi(1≤i≤r);λ2r+1,···,λn

(2)A∈Mn(R) (ATA=AAT) A

diag a1 b1

,···, ar br

,λ ,···,λ

−b1 a1

−br ar

2r+1 n

(1)

bi>0(1≤i≤r)A Æ DZ

√

ai±−1bi(1≤i≤r);λ2r+1,···,λn

( 13.25)

AV(~b1,···,~bn)Ædiag(λ1,···,λn)

A(~bi)=λi~bi(1≤i≤n)

14.41≤i≤nA∗(~bi)=λi~bi 1≤i≤n

(AA∗−A∗A)(~bi)=~0

AA∗=A∗A

A(V,hi)λi∈Spec(A)

Wi={~x∈V|A(~x)=λi~x}

14.5WiDZA∗

W⊥ADZA∗

i

i

~y∈W⊥ x~∈Wi

∗

hA(y)|~xi=hy~|A(x~)i=0

∗∗ ⊥ ∗

(A)=AWiDZA

i

i

AA∗W⊥AW⊥DZ

i

i

Wi⊕=VAW⊥Æ V=Wi⊕W⊥

14.7(Æ). (1)A∈L(V)(V,hi)

···

AVDZÆ

diag a1 b1

−b1 a1

, , ar br

−br ar

,λ2r+1,···,λn

bi>0(1≤i≤r)AÆDZ

ai±√−1bi(1≤i≤r);λ2r+1,···,λn

···

A∈Mn(R)(ATA=AAT)A

diag a1 b1

−b1 a1

, , ar br

−br ar

,λ2r+1,···,λn

bi>0(1≤i≤r)AÆDZ

ai±√−1bi(1≤i≤r);λ2r+1,···,λn

(1)

(13.25)

154

( ). F=R

V

A:V→V (V,hi)

V=W1⊕W2⊕···⊕Wm

Wi 2AA∗ (Wi AA∗ )

i=⊕jWi⊂W⊥j

DZ

dim(Wi)=2 AA∗ Wi (~u1,~u2)

λ0i μ0i ; λ0i −μ0i ;

−μ0i λ0i

μ0i

λ0i

√

λ0i±μ0i−1A Æ λ0i,μ0i∈R,μ0i⊕=0

V=Rnh~x|~yi=ρn(~x,~y)=~xT·~y

λ1A Æ ~uA λ1 Æ

W1={α~u|α∈R}

1

W1 W⊥ A A∗

W

DZdim(W1)=1 W1AA∗ AA∗W⊥1

⊥

1

n=2 AR2 Æ A Æ B=(~e1,~e2)DZR2

A DZ

a b

A= cd

A A ATA=AATA

a2+b2=a2+c2,ac+bd=ab+cd,c2+d2=b2+d2

b2=c2 (a−d)(c−b)=0

Æ |λE2−A|=λ2−(a+d)λ+(ad−bc)

(a+b)2−4(ad−bc)=(a−d)2+4bc<0

bc<0 a=dc=−b=⊕0

A ÆDZa±

A= a b

−b a

√

−1|b|(b=⊕0) b<0 H12AH12= a |b|

−|b| a

n>2 ARn Æ A Æ

C:Rn→Rn DZCn C:Cn→Cn

∀~x,~y∈RnC(~x+√−1~y)=C(~x)+√−1C(~y)

∗

C =C∗

B◦C=B◦C

∗ ∗ ∗ ∗

CC=CC CC =C C

A → hi

14.8().F=R :V V(V, )

V

V=W1⊕W2⊕···⊕Wm

Wi2AA∗(WiAA∗)

j

i⊕=jWi⊂W⊥dim(Wi)=2AA∗ Wi (~u1,~u2)

μ λ ;

DZ

λ0i μ0i

−μ0i λ0i

; λ0i −μ0i

0i 0i

λ0i±μ0i√−1AÆλ0i,μ0i∈R,μ0i⊕=0

V=Rnh~x|~yi=ρn(~x,~y)=~xT·~y

λ1AÆ~uAλ1Æ

W1={α~u|α∈R}

1

W1W⊥AA∗

1

DZdim(W1)=1W1AA∗AA∗W⊥

1

W⊥

n=2AR2ÆAÆB=(~e1,~e2)DZR2

ADZ

a b

A= c d

AAATA=AAT

A

a2+b2=a2+c2,ac+bd=ab+cd,c2+d2=b2+d2

b2=c2(a−d)(c−b)=0

AÆ|λE2−A|=λ2−(a+d)λ+(ad−bc)

(a+b)2−4(ad−bc)=(a−d)2+4bc<0

bc<0a=dc=−b⊕=0

A= a b

−ba

AÆDZa±√−1|b|(b⊕=0)b<0 H12AH12=

−|b| a

a |b|

n>2ARnÆAÆ

√

√

C:Rn→RnDZCnC:Cn→Cn

n

∀~x,~y∈R

C(~x+

−1~y)=C(~x)+

−1C(~y)

∗ ∗

C =C

B◦C=B◦C

∗ ∗ ∗ ∗

CC=CCCC =C

C

155

( )

√Cn A Æλ0+μ0

√

−1(λ0,μ0∈R,μ0⊕=0) Æ Æ

~u=~u1+

–1~u2

~u1,~u2∈Rn

A∗(~u1)=λ0~u1−μ0~u2,A(∗~u2)=μ0~u1+λ0~u2

A(~u1)=λ0~u1+μ0~u2,A(~u2)=−μ0~u1+λ0~u2

~u1,~u2 ~u1=r~u2r∈R

√ √ √ √ √ √

~u1+−1~u2=(r+ −1)~u2;(r+ −1)A(~u2)=A((r+ −1)~u2)=(λ0+μ0−1)(r+ −1)~u2

√

A(~u2)=(λ0+μ0−1)~u2 DZA(~u2)∈Rn

W1=h{~u1,~u2}i

dim(W1)=2W1AA∗

1

W⊥DZAA∗

(ii) W⊥={~1x∈Rn|h~x|~u1i=h~x|~u2i=0}

1

~x∈W⊥ A(~x)∈W⊥1

A∗(~x)∈W⊥1

hA(~x)|~u1i=h~x|A∗(~u1)i=0;hA(~x)|~u2i=h~x|A∗(~u2)i=0;

1

DZA∗(~u1)∈W1 A∗(~u2)∈W1 ~x∈W⊥

hA∗(~x)|~u1i=h~x|A(~u1)i=0;hA∗(~x)|~u2i=h~x|A(~u2)i=0;

DZA∗∗=A

V=W1⊕W⊥W1AA∗ (2) W⊥DZAA∗

DZDZn−2AA∗W⊥1 A W⊥ DZ1 Æ

1 1

W1⊥

14.1.3

14.9. DZn ()Æ

⎛A11 A12 ··· A1k B11 ··· B1r

⎞⎛ ⎞

A11 O2 ··· O2 O21 ··· O21

⎜O2 A22 ···

A2k B21 ···

B

2r⎟

A21 A22 ··· O2 O21 ··· O21⎟

⎜

⎜ . . . . . . . ⎟⎜

. ⎟

⎜O2 O2 ··· Akk Bk1 ··· Bkr

, Ak1 Ak2 ··· Akk O21 ··· O21⎟

O12 O12 ··· O12 λ11 ··· λ1r⎟⎜C11 C12 ··· C1k λ11 ··· 0

⎜ ⎟⎜ ⎟

⎝ . . . . . . . ⎠⎝ ⎠

O12 O12

···

O12

0

···

λrr

Cr1 Cr2

···

Crk

λr1

···

λrr

0≤r≤nr+2k=nk1≤j≤rBij2×1

1≤i,j≤r

Cji1×2

λij∈R

O22×2

1≤i,j≤kAij2×2

O212×1

O12

1≤i≤

1×2

A

Æ

A

AT

Æ

14.10(

B

Æ

).

Æ

(1)n

A

(V,hi)

Æ

Æ

A

V

A∈Mn(R) Æ A Æ Æ

()

~u=~u+√1~u ~u,~u Rn

CnAÆλ0+μ0√−1(λ0,μ0∈R,μ0⊕=0)ÆÆ

1 − 2 1 2∈

A(~u1)=λ0~u1−μ0~u2,A(~u2)=μ0~u1+λ0~u2

∗ ∗

A(~u1)=λ0~u1+μ0~u2,A(~u2)=−μ0~u1+λ0~u2

~u1,~u2~u1=r~u2r∈R

~u1+√−1~u2=(r+√−1)~u2;(r+√−1)A(~u2)=A((r+√−1)~u2)=(λ0+μ0√−1)(r+√−1)~u2

A(~u2)=(λ0+μ0√−1)~u2DZA(~u2)∈Rn

W1=h{~u1,~u2}i

dim(W1)=2W1AA∗

1

W⊥DZAA∗

1

(ii)W⊥={~x∈Rn|h~x|~u1i=h~x|~u2i=0}

1

1

1

~x∈W⊥ A(~x)∈W⊥A∗(~x)∈W⊥

∗ ∗

hA(~x)|~u1i=h~x|A(~u1)i=0;hA(~x)|~u2i=h~x|A(~u2)i=0;

1

DZA∗(~u1)∈W1A∗(~u2)∈W1~x∈W⊥

∗ ∗

hA(~x)|~u1i=h~x|A(~u1)i=0;hA(~x)|~u2i=h~x|A(~u2)i=0;

DZA∗∗=A

1

1

V=W1⊕W⊥W1AA∗(2)W⊥DZAA∗

DZDZn−2AA∗W⊥AW⊥DZÆ

1 1

1

W⊥

14.9.DZn()Æ

⎛

O A A B B ⎞

A11 A12 ··· A1k B11 ··· B1r

⎜

⎟

⎜ 2 22 ··· 2k 21 ··· 2r⎟

.

.

.

.

.

.

.

A11 O2 ··· O2 O21 ··· O21A21 A22 ··· O2 O21 ··· O21

⎜

.

.

.

.

.

.

.

⎟

⎛

⎞

⎟

⎜

, Ak1 k2 ··· Akk 21 ··· O21⎟

⎜O2 O2 ··· Akk Bk1 ··· Bkr⎟⎜ A O

O12 O12 ··· O12 λ11 ··· λ1r

⎜ ⎟

C11 C12 ··· C1k λ11 ··· 0

⎜ ⎟

O12 O12 ··· O12 0 ··· λrr

Cr1 Cr2 ··· Crk λr1 ··· λrr

⎝ . . . . . . . ⎠⎝ ⎠

0≤r≤nr+2k=n1≤i,j≤rλij∈R1≤i,j≤kAij2×21≤i≤

k1≤j≤rBij2×1Cji1×2O22×2O212×1O121×2

AÆAATÆ

14.10(Æ). (1)n(V,hi) AV

BÆAÆÆ

A∈Mn(R)ÆAÆÆ

156

n

(2)

(2) (1)

(1)

n=2

∈

A

M2(R)

Æ

A,AT

( )Æ

A

Æ ~u ~v ~u1,~u2 AAT A

Æ Æ A Æλ1 ~u1 A

λ1 Æ ~vDZ ~u1 R2

P=(~u1,~u2) P

{~u1,}~v ~u1 ~u2

AP=P λ1 α

0 μ

|λE2−A|=(λ−λ1)2=(λ−λ1)(λ−μ)μ=λ1 A Æ

AP=P λ1 α

λ1

;ATP=P λ1 0

α λ1

n>2 n A ∈Mn(R) A Æ

A Æ

λnDZA Æ ~u∈n RnDZA λn Æ ~un Rn

(·~·u·n,~b1, ,~bn−2,~bn−1) DZ P1 P2=P1H1n

~un P2 n

P2=~bn−1,~b1,···,~bn−2,~un=(~u1,~u2,···,~un−1,~un)

2

PTAP2 ⎛ T T ⎞

⎜

~u1A~u1···~u1A~un

⎠

PTAP2=⎝ ⎟

2

≤i≤n~uTA~un=λn~uT~un=λnδin

i i

. ··· .

~uTA~u1···~uTA~un

n n

2 λ

PTAP2= A1 O1

A2 n

⎞

A1 (n−1)×(n−1) A2 1×(n−1) O1 (n−1)×1(n−1) Q QTA1QDZ Æ

⎛A11 O2 ··· O2 O21 ···O21

A

⎜ 21

A22 ···O2

O21 ···

O

21⎟

⎜ ⎟

QTA1Q=⎜Ak1 Ak2 ···Akk O21 ···O21⎟

⎜C11 C12 ···C1k λ11 ··· 0 ⎟

⎝ ⎠

Cr1 Cr2 ···Crk λr1 λrr

Aii(1≤i≤k) Æ 2k+r=n−1

P=P2

Q O1 ,

O2 1

n(2)(2)(1)(1)

∈

n=2A M2(R)ÆA,AT()ÆA

Æ~u~v~u1,~u2AATA

ÆÆAÆλ1~u1A

{ }

λ1Æ~vDZ~u1R2~u1,~v~u1~u2

P=(~u1,~u2)P

AP=P λ1 α

0 μ

|λE2−A|=(λ−λ1)2=(λ−λ1)(λ−μ)μ=λ1AÆ

AP=P λ1 α

0 λ1

;ATP=P λ1 0

α λ1

∈

n>2nA Mn(R)AÆ

AÆ

∈

λnDZAÆ~un RnDZAλnÆ~unRn

···

(~un,~b1, ,~bn−2,~bn−1)DZP1P2=P1H1n

~unP2n

P2=~bn−1,~b1,···,~bn−2,~un=(~u1,~u2,···,~un−1,~un)

2

PTAP2

⎛ T T ⎞

2

~u1A~u1 ··· ~u1A~un

PTAP2=⎜⎝

. ···

. ⎟⎠

~uTA~u1 ··· ~uTA~un

n n

1≤i≤n~uTA~un=λn~uT~un=λnδin

i i

2

A2

λn

PTAP2=A1 O1

A1(n−1)×(n−1)A21×(n−1)O1(n−1)×1

⎛

A A O O O ⎞

(n−1)QQTA1QDZÆ

A11 O2 ··· O2 O21 ··· O21

⎜

.

.

.

.

.

.

.

⎟

⎜ 21 22 ··· 2 21 ··· 21⎟

QTA1Q=⎜Ak1 Ak2 ··· Akk O21 ··· O21⎟

C11 C12 ··· C1k λ11 ··· 0

⎜ ⎟

⎝ ⎠

Cr1 Cr2 ··· Crk λr1 ··· λrr

2

Aii(1≤i≤k)Æ2k+r=n−1

P=P Q O1 ,O2 1

157

⎞

O1 (n−1)×1 O2 1×(n−1) P

⎛A11 O2 ··· O2 O21 ···O21 O21

A

⎜ 21

A22 ···O2

O21 ···O21

O

21⎟

⎜ . . . . . . . ⎟

PTAP=⎜Ak1 Ak2 ···Akk O21 ···O21 O21⎟

C11

⎜

C12 ···C1k λ11 ··· 0 0 ⎟

⎟

. . . . . . . .

⎜ ⎟

Cr1d1

Cr2d2

···

···

Crkdk

λr1

e1

···λrr

··· er

0

λn

A2Q

P

PT=P−1

⎝ ⎠

(d1,d2,···,dk,e1,···,e1r)= A Æ

DZλii(1≤i≤r),λnA Æ Aii(1≤i≤k)A Æ √

A Cn √ Cn A Æλ0+μ0 −1 λ0,μ0 μ0

=⊕0

~u∈CnDZA λ0+μ0

~u1,~u2∈Rn

−1 Æ ~u

~u=~u1+

√

−1~u2

√ √ √ √

A(~u1+ −1~u2)=(λ0+μ0−1)(~u1+ −1~u2)=(λ0~u1−μ0~u2)+ −1(λ0~u1+μ0~u2)

A(~u,~u)=(~u,~u) λ0 μ0

1 2 1 2

−μ0 λ0

Rn ~u1,~u2 ~u1=r~u2r∈R

√ √ √ √ √ √

~u1+

−1~u2=(r+ −1)~u2;(r+−1)A~u2=A((r+−1)u2~)=(λ0+μ0−1)(r+ −1)~u2

√

A~u2=(λ0+μ0−1)~u2 DZA~u2∈Rn

~u1,~u2 Rn (~u1,~u2,~u3,···,~un) (

)(~bn,~bn−1,···,~b1)

~bn−1

= 1~u

k~u1k

1=b~u1,

~bn=c~u1+d~u2,c∈R,d>0

bd

~u1=b−1~bn−1,~u2=d−1~bn c~bn−1;~bT~u1=~bT~u2=0(1≤i≤n−2)

— i i

A(~bn−1,~bn)=(A~bn−1,A~bn)=(bA~u1,cA~u1+dA~u2)

−1

bc

λ

μ0

bc

0d

0

−μ0

λ0

0d

A(~b ,~b)=(~b,~b)

n−1n n−1 n

1≤i≤n−2

~biTA~bn−1=b~biTA~u1=b~ibT(λ0~u1−μ0~u2)=0,

~bTA~bn=c~bTA~u1+d~bTA~u2=d~bT(μ0~u1+λ0~u2)=0

i i i i

P1=~b1,~b2,···,~bn−1,~bn

⎛

A A O O O O ⎞

O1(n−1)×1O21×(n−1)P

A11 O2 ··· O2 O21 ··· O21 O21

⎜

⎟

⎜ 21 22 ··· 2 21 ··· 21 21⎟

.

.

.

.

.

.

.

⎟

PTAP=⎜Ak1 Ak2 ··· Akk O21 ··· O21 O21⎟

C11

.

.

.

.

.

.

.

.

⎜

C12

··· C1k

λ11

··· 0 0

⎟

Cr1 Cr2 ··· Crk λr1 ··· λrr 0

d1 d2 ··· dk e1 ··· er λn

⎜⎝ ⎟⎠

(d1,d2,···,dk,e1,···,e1r)=A2Q PPT=P−1AÆ

DZλii(1≤i≤r),λnAÆAii(1≤i≤k)

√

AÆ

ACn√CnAÆλ0+μ0

−1λ0,μ0μ0⊕=0

~u∈CnDZAλ0+μ0

~u1,~u2∈Rn

−1Æ~u

~u=~u1+√−1~u2

A(~u1+√−1~u2)=(λ0+μ0√−1)(~u1+√−1~u2)=(λ0~u1−μ0~u2)+√−1(λ0~u1+μ0~u2)

1 2 1 2

A(~u,~u)=(~u,~u) λ0 μ0

−μ0 λ0

Rn~u1,~u2~u1=r~u2r∈R

~u1+√−1~u2=(r+√−1)~u2;(r+√−1)A~u2=A((r+√−1)~u2)=(λ0+μ0√−1)(r+√−1)~u2

A~u2=(λ0+μ0√−1)~u2 DZA~u2∈Rn

~u1,~u2Rn(~u1,~u2,~u3,···,~un)(

)(~bn,~bn−1,···,~b1)

~bn−1

1

= ~u1

k~u1k

=b~u1,~bn

=c~u1

+d~u2,c∈R,d>0

~u1

=b−1~b

n−1

,~u2

=d−1~bn

c~bbd

n−1

;~bT~u1

=~bT~u2

=0(1≤i≤n−2)

—

i

i

A(~bn−1,~bn)=(A~bn−1,A~bn)=(bA~u1,cA~u1+dA~u2)

A(~bn−1,~bn)=(~bn−1,~bn)

b c

−1λ

μ0b c

0

0d −μ0 λ0 0d

1≤i≤n−2

i

i

i

~bTA~bn−1=b~bTA~u1=b~bT(λ0~u1−μ0~u2)=0,

~bTA~bn=c~bTA~u1+d~bTA~u2=d~bT(μ0~u1+λ0~u2)=0

i i i i

P1=~b1,~b2,···,~bn−1,~bn

⎜

1

.

.

1

.

⎟

=

⎜

~bT

A~b

~bT

A~b

⎟

⎛~bTA~b1 ··· ~bTA~bn⎞

⎛A1 O1 ⎞

PTAP

=

~T

~

bnAb1 ···

1 1 ⎝ . .

~T

~

. ⎠ ⎝A2

bnAbn

n−1

n

n−1

n−1 n ⎠

n

~bTA~bn−1 ~bTA~bn

158

⎛~ ~

~ ~⎞ ⎛A O ⎞

⎜1 .

1

bTAb1···

.

bTAbn

.

1

⎟=⎜

~bT

A~b

1

~bT A~b ⎟

1

PTAP1=⎝

. . .

⎠ ⎝A2 n−1

n−1

n−1 n ⎠

~bTAb~ ··· ~bTA~bn n n

n 1 n

~bTA~bn−1 ~bTA~bn

⎛

A1(n−2)×(n−2)O1(n−2)×2A22×(n−2)

A1 O1

T −1

⎞ A1 O1

2

P1AP1=⎝A

b c

0 d

B=b c

λ0 μ0

⎠

−μ0 λ0

0

0

−1λ μ

b c =

0 d

b c

A2 B

0 d −μ0 λ0 0 d

A1ÆDZAÆ

A1DZÆBλ0 μ0

√ √

−μ0 λ0

T

Æλ0+μ0−1λ0−μ0−1Q1DZ(n−2)Q1A1Q1DZ

2

ÆD1

P= Q1 O1O2 E2

O1(n−2)×2O22×(n−2)E22P2DZn-

AP=P2P1AP1P2=

1

A2Q1 B

=

P=P1P2P

P

T TT

QTA1Q1 O1

D1 O1

A2Q1 B

···

14.11.A∈Mn(R)(ATA=AAT)A

diag a1 b1

−b1 a1

bi>0(1≤i≤r)AÆDZ

, , ar br

−br ar

,λ2r+1,···,λn

ai±√−1bi(1≤i≤r);λ2r+1,···,λn

∈

A Mn(R)Æ(14.10)PB=(bij)

ÆBÆDZ

ci di

D=diag(A1,A2,···,Ar,λ2r+1,···,λn);Ai=ai bi,(1≤i≤r)

tr(BB)=b

n

T 2

ij

i,j=1

tr(BB)≥tr(AAi)+

n

λ

r

T T 2

i i

i=1 k=2r+1