1 12

8

8.1

1 11 10 0

A =

, [ ]2 1 2B = , 1 0 11 1 1 = .

: y A

( ) { }T T 12 2 4, 0 4 22 2A A A A = = = = . , ( 1 2 = 1 4 = ),

[ ]T1 1 1 12x =

TA A

[ T1 11

1 1 1 1 02

y Ax = = ] . TAA, TA Ax 0= [ T2 1 1 12x = ] TAA y 0=

: : 5, . 191

: 1, 3, 4, 8, 9, . 198.

2 12

[ ] [ ]T T1 2c 1 1 0 c 0 0 1y = + . [ T2 1 1 1 02 ]y = [ ]

T3 0 0 1y =

. , ( Tker AA )

1 111 12

V = ,

1 1 02 2

1 1 02 2

0 0

U

1

=

#

#

#

,

2 00 0

0 0

AS

= " "

T

AA US V= . ,

U S ,

A

A

T T

1 1 1 12 2 2 2

2 0 2 01 1 1 12 2 2 20 0 0 0

0 0 0 0

A V V V

= =

TV .

T2 0 1 10 0 1 1

P V V = = ,

T

1 12 2 1 0

1 1 0 12 2

0 00 0

Q V

= =

. A QP= A

y B [ ]T 9BB = 1 9 3 = = . [ ]3 0 0BS = [ ]1U = , [ ]1 1y = , TBB 1 3 = ,

. 1 9 =

3 12

, T

T1 1

1 2 1 23 3 33

x B y = =

. T 9B Bx x=

T

4 2 42 1 24 2 4

B Bx 0 x 0 = =

[ ] [ ]T1 2c 1 2 0 c 0 2 1x = + T . [ ]T1 2 0 [ ]T0 2 1 , Gram-Schmidt

[ ]T2 1 1 2 05x = , [ ]T

31 4 2 5

3 5x =

( )Tker B B . ,

[ ][ ]

T2 1 4

3 5 3 51 2 21 3 0 0 3 5 3 5

52 03 3

B

=

#

[ ][ ][ ] 2 1 21 3 1 3 3 3B = = PQ , [ ][ ][ ] [ ]1 3 1 3 0P = = > [ ] 2 1 2 2 1 21 3 3 3 3 3 3Q = = . [ ]T 1QQ = . y

( ) { }T T 1 22 0 2, 3 2, 30 3 = = = =

2 0 0

0 3 0S

= .

4 12

T 1 2[ ]T1 1 0y = [ ]T2 0 1y = . ( ) { }Tker 0= , [ ]1 2U y y I= = 2 .

[ ]TT1 11 1 1 0 12 2x y = = , [ ]TT

2 21 1 1 1 13 3

x y = =

T [ T3 1 1 2 16x = ])

(ker T . , [ ]1 2 3V x x x= T

2

1 1 12 3 6

2 0 0 1 203 60 3 0

1 1 12 3 6

I

=

##

1 102 0 2 21 1 20 3

3 3 6

PQ = =

,

, T 2QQ I= .

* * *

8.2 , ^ , A = . : , rankA 1A = .

2 2 1A A = = = . A USV=

( )diag , 0, , 0S = 2V x x =

,

2 , ,x x ( ) { }ker spanA A = .

5 12

22 2

1

1 1 1 y A A

= = = =

AA

2U y

y =

2 , ,y y ( ) { }ker spanAA = .

( )( )A USV USU UV PQ = = =

( )( )A USV UV VSV QP = = = .

* * *

8.3 , TAX XA=4 2 24 4 12 4 2

A =

.

: ,

TA USV=

1 4 235 3 5

2 2 135 3 5

5 20 33 5

U

=

,

2 1 05 5

1 2 05 5

0 0

V

1

=

( )diag 6, 6, 9S = . TAX XA=

( ) ( )T T T TUSV X XVSU S V XU U XV S SY Y S= = = T ,

. TY V XU= Y

6 12

T11 122 2 T11 11Y Y=11 12T T T12 22 12 22

6 69 9

Y YI 0 I 0Y YY Y0 0Y Y

= , T22 22Y Y= 12Y O=

( ) T1 2diag ,X V Y Y U= , 21,Y Y2 2 1 . 1

* * *

8.4 A

1 2det A = " . : ( )rank rankAA A = = , AA

( ) 2 2 21 2det AA = " , ( )2i i AA = . ,

( ) ( )( ) ( )( ) 2det det det det det detAA A A A A A = = = 1 2det A = " .

* * *

8.5

,

1 1 2 2A P Q Q P= =A 1 2Q Q= .

: TA USV=( )( )T T T 1 1A USV USU UV P Q= = =

( )( )T T T 2 2A USV UV VSV Q P= = = . T1 2Q Q UV= =

* * *

7 12

8.6 1 2, , , " ,

A \

1 2tr A + + +" . : ,

TA USV=TQ V U=

( ) ( )T TA V V US V V QS V= = T . Q , ijq ijq 1

( )( ) ( )( ) ( )T T 11 1 22 21 11 2 22 1 2

tr tr tr tr q q q

q q q .

A V QS V QS V V QS

= = = = + + + + + + + + +

"" "

* * *

8.7 1 2, , , " .

A ^

. 1 11 2, , , 1 1A . . . 1 2 1 = = = =" A : .

( )( ) ( )( ) ( )1 11 1 1A A A A A A = = i ( )( )1 1A A ,

. ,

1i i

=

(i A A ) 1A 1A{ } { }1 1i i

i

1A

= = = ,

. i A. 1 2 1 A UV = = = = =" , S I A= .

8 12

, , A

i 1 1, i 1,2, ,AA A A I = = = = = .

* * *

8.8

A

. ( )1 2diag , , ,A V V= . . A PQ= , .

,P Q

: . A , x ,

A

Ax x= A x x = . 2AA x Ax x = = 2A Ax A x x = = .

, A U V= ( )1 2diag , , ,S = .

, ( )1 2diag , , ,A V V= ( )22 21 2diag , , ,AA V V A A = = .

. , . A PQ= P QQ I = A

( ) 12 2 2 2AA A A P Q P Q P P Q PQ QP PQ = = = = = . , QP PQ PQ Q P = = ,

A

2AA PQQ P P = = , ( ) ( ) 2A A Q P PQ PQ QP P = = = .

* * *

9 12

8.9 ,A B \ , . A MBN= , .

M N

. , . T TAA BB A BT= = T : . , .

,

A MBN= M NTAA TBB

( )T TAA M BB M= T

2

,

, . TAA TBB

,

,

T1 1 1A U S V = T2 2 2B U S V =

A B 1S S= , ( ) ( )T T1 2 2 1A U U B V V MBN= = ,

, . M N

.

TAA BB= T A Biy . ,

,A BT

1A USV= , T2B USV= , . ( )T2 1A B V V BT= = T2 1T V V=, , , A BT= T

T T TAA BTT B BBT= = .

* * *

8.10 A \ Az b= ,

( )Tk ii2

i 1 i

A b xz x

=

= D

, k rank A= , i .

A

ixTA A

10 12

: [ ]TT 1 2 1 2A USV y y y S x x x = = , A

( )1 2 kdiag , , , OS

O O

=

#""""""""" # "

#

Ax b= S = ,

, T V z= T U b= .

( )1 2 k1 1

2 2

diag , , , O

O O

=

#""""""""" # "

#.

2 0= , 2 0 . ,

T

11 12 1k1 1

1 2 k 1 2 k

1 1 1diag , , , = =

"

,

( ) ( )TT ii1i i i i2 2 2

i i i i i

A b xb Axy b b y = = = = DDD .

( )Tk k i1 1ii i2

i 1 i 1i i

A b xz V V x

0 = =

= = = = D x .

* * *

8.11 ( )max ( )min . :

. , x ( ) ( )min max2 2A x Ax A x 2

. ( ) ( ) ( )max max maxA B A B + +

11 12

: . A USV= .

A

( )1 2 kdiag s , s , , s , 0, , 0S = ( ) k2 22 2 2i2 i2 i 1Ax x A Ax x VSU USV x S V x V x == = = = .

V

( ) ( )22 22 2max max2 22Ax A V x A x =

( ) ( )22 22 2min min2 22Ax A V x A x = . . ( )max2A A= , ( )max2B = B

( ) ( ) ( )max max max2 2 2A B A B A B A B + = + + = + .

* * *

8.12 ,

,

1 2 k, , , " A ^k rank A =

( ) ( + + )O A

BA O = .

: , A USV= A2 AA OB

O A A

= .

, ( ) ( ) ( ) { }2 2 21 2 k, , , , 0B AA A A = = 2 ( ) { }1 2 k, , , , 0B = .

2i iA Ax x = i i2i iAA y y = , ( 7.25, . 192 )

i i i ii

i i i i

i

i

y Ax y yB

x A y x x = = = ,

12 12

i i i ii

i i i i

i

i

y Ax y yB

x A y x x = = = ,

i , ii

yx i ,

i

i

yx

, i 1 . B , 2, , k= 0 = B( ) ( ) ( )rank 2rank 2kB A+ = + = +

. ( )( )i ker ker A A A = ( )( )i ker kerz AA A = , 0 = B

i

0 , ( )i 1,2, , k=

jz0

, ( )j 1,2, , k= ,

i

i

0 AB 0

0 =

= , jj

0zB 0

A z0 = =

.

* * *