14

14.1

( ) ? ? ?

? ? ? ( ) ? ( ) ? ( )

?

14.1.1

( ). (V,hi)? V A ( )

A??A=A?A?

( ). A∈Mn(C)∪Mn(R)A ( ) A?·A=A·A?

A A

A=diag(2i,2,1,···,1)

?

(~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)=~0? A?(~bi)=λi~bi

(1)A A?λE(λ∈C)

(2) A λ∈C ~x∈VA(~x)=λ~x??A?(~x)=λ~x

?(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

( ? ). (1) (V,hi) AV

? A?V

(2)A∈Mn(C) ? A

()???

???()?()?()

?

14.1().(V,hi)?VA()

?

?

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)

?

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)=~0?A?(~bi)=λi~bi

14.5. (1)AA?λE(λ∈C)

(2)Aλ∈C~x∈VA(~x)=λ~x??A?(~x)=λ~x

(1)?(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(?). (1)(V,hi)AV

?A?V

(2)A∈Mn(C)?A

153

AV (~b1,···,~bn) ? 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}

?

...............................................................................................................................................................................................................................................................................................................................................................................................................................14.5Wi?A

W⊥ A ?A?

i

i

~y∈W⊥ x~∈Wi

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

??

⊥

?

(A)=A

Wi?

A

i i

A A? W⊥ A W⊥ ?

Wi=⊕V A W⊥i ? V=Wi⊕W⊥i

( ? ). (1)A∈L(V) (V,hi)A V ? ?

diag a1 b1

,···, ar br

,λ ,···,λ

?b1 a1

?br ar

2r+1 n

bi>0(1≤i≤r)A ? ?

√

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 ? ?

√

ai±?1bi(1≤i≤r);λ2r+1,···,λn

( 13.25)

AV(~b1,···,~bn)?diag(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}

14.5Wi?A?

W⊥A?A?

i

i

~y∈W⊥ x~∈Wi

?

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

?? ⊥ ?

(A)=AWi?A

i

i

AA?W⊥AW⊥?

i

i

Wi⊕=VAW⊥? V=Wi⊕W⊥

14.7(?). (1)A∈L(V)(V,hi)

···

AV??

diag a1 b1

?b1 a1

, , ar br

?br ar

,λ2r+1,···,λn

bi>0(1≤i≤r)A??

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)A??

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

?

dim(Wi)=2 AA? Wi (~u1,~u2)

λ0i μ0i ; λ0i ?μ0i ;

?μ0i λ0i

μ0i

λ0i

√

λ0i±μ0i?1A ? λ0i,μ0i∈R,μ0i⊕=0

V=Rnh~x|~yi=ρn(~x,~y)=~xT·~y

λ1A ? ~uA λ1 ?

W1={α~u|α∈R}

1

W1 W⊥ A A?

W

?dim(W1)=1 W1AA? AA?W⊥1

⊥

1

n=2 AR2 ? A ? B=(~e1,~e2)?R2

A ?

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

? |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 ??a±

A= a b

?b a

√

?1|b|(b=⊕0) b<0 H12AH12= a |b|

?|b| a

n>2 ARn ? A ?

C:Rn→Rn ?Cn 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 ;

?

λ0i μ0i

?μ0i λ0i

; λ0i ?μ0i

0i 0i

λ0i±μ0i√?1A?λ0i,μ0i∈R,μ0i⊕=0

V=Rnh~x|~yi=ρn(~x,~y)=~xT·~y

λ1A?~uAλ1?

W1={α~u|α∈R}

1

W1W⊥AA?

1

?dim(W1)=1W1AA?AA?W⊥

1

W⊥

n=2AR2?A?B=(~e1,~e2)?R2

A?

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

A?|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

A??a±√?1|b|(b⊕=0)b<0 H12AH12=

?|b| a

a |b|

n>2ARn?A?

√

√

C:Rn→Rn?CnC: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 ?λ0+μ0

√

?1(λ0,μ0∈R,μ0⊕=0) ? ?

~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 ?A(~u2)∈Rn

W1=h{~u1,~u2}i

dim(W1)=2W1AA?

1

W⊥?AA?

(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

?A?(~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;

?A??=A

V=W1⊕W⊥W1AA? (2) W⊥?AA?

??n?2AA?W⊥1 A W⊥ ?1 ?

1 1

W1⊥

14.1.3

14.9. ?n ()?

?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

?

A

AT

?

14.10(

B

?

).

?

(1)n

A

(V,hi)

?

?

A

V

A∈Mn(R) ? A ? ?

()

~u=~u+√1~u ~u,~u Rn

CnA?λ0+μ0√?1(λ0,μ0∈R,μ0⊕=0)??

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)~u2?A(~u2)∈Rn

W1=h{~u1,~u2}i

dim(W1)=2W1AA?

1

W⊥?AA?

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

?A?(~u1)∈W1A?(~u2)∈W1~x∈W⊥

? ?

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

?A??=A

1

1

V=W1⊕W⊥W1AA?(2)W⊥?AA?

??n?2AA?W⊥AW⊥??

1 1

1

W⊥

14.9.?n()?

?

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

A?AAT?

14.10(?). (1)n(V,hi) AV

B?A??

A∈Mn(R)?A??

156

n

(2)

(2) (1)

(1)

n=2

∈

A

M2(R)

?

A,AT

( )?

A

? ~u ~v ~u1,~u2 AAT A

? ? A ?λ1 ~u1 A

λ1 ? ~v? ~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 ?

AP=P λ1 α

λ1

;ATP=P λ1 0

α λ1

n>2 n A ∈Mn(R) A ?

A ?

λn?A ? ~u∈n Rn?A λn ? ~un Rn

(·~·u·n,~b1, ,~bn?2,~bn?1) ? 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 QTA1Q? ?

?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) ? 2k+r=n?1

P=P2

Q O1 ,

O2 1

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

∈

n=2A M2(R)?A,AT()?A

?~u~v~u1,~u2AATA

??A?λ1~u1A

{ }

λ1?~v?~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=λ1A?

AP=P λ1 α

0 λ1

;ATP=P λ1 0

α λ1

∈

n>2nA Mn(R)A?

A?

∈

λn?A?~un Rn?Aλn?~unRn

···

(~un,~b1, ,~bn?2,~bn?1)?P1P2=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)QQTA1Q??

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)?2k+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 ?

?λii(1≤i≤r),λnA ? Aii(1≤i≤k)A ? √

A Cn √ Cn A ?λ0+μ0 ?1 λ0,μ0 μ0

=⊕0

~u∈Cn?A λ0+μ0

~u1,~u2∈Rn

?1 ? ~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 ?A~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?1A?

?λii(1≤i≤r),λnA?Aii(1≤i≤k)

√

A?

ACn√CnA?λ0+μ0

?1λ0,μ0μ0⊕=0

~u∈Cn?Aλ0+μ0

~u1,~u2∈Rn

?1?~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 ?A~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

A1??A?

A1??Bλ0 μ0

√ √

?μ0 λ0

T

?λ0+μ0?1λ0?μ0?1Q1?(n?2)Q1A1Q1?

2

?D1

P= Q1 O1O2 E2

O1(n?2)×2O22×(n?2)E22P2?n-

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)A??

, , ar br

?br ar

,λ2r+1,···,λn

ai±√?1bi(1≤i≤r);λ2r+1,···,λn

∈

A Mn(R)?(14.10)PB=(bij)

?B??

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