1 13

7

CAYLEY-HAMILTON

: : 6, . 160

: 1, 2, 3, . 162.

. Cayley-Hamilton A p( ) , ( )p A O= , .

(7.1) 11 1( ) c c c = + + + +" 0 ( )A O = , . A :

1. , ( )diag , , ,A I= = ( )A I O = = .

2. , 1 2 20 3 40 2 3

A =

I2A = , A I

2( ) 1 = .

3. , 1 2 22 1 22 2 1

B =

I2 4 5B B= + ,

2( ) 4 5 = . 4. ( 2A A= ) 2( ) = .

2 13

5. ( )A O = . ( ) =

7.1 ( ) A( ) , A ( ) ( ) .

: ( ) ( )( ) ( ) ,

( ) ( ) ( ) ( ) = + . ( ) (( ) ( ) < )

( ) ( ) ( ) ( ) ( )A A A A A = + = O , , A ( ) .

( ) A , p( ) p( )A O= .

7.2 . A

: (7.1) ( ) A1

1 1 0( ) d d d

= + + + +" , ( )A O = .

( ) ( )11 1 1 1 0( ) ( ) ( ) c d c d c d = = + + + " 0

( ) ( ) ( )A A A = = O , , ( )( ) 1 = ( ) . A

7.3 A

. ( ) A

3 13

: 0 A0( ) ( ) ( ) c = + , c 0 .

0( ) ( ) ( ) cA A I A I = + = O

0( )( )cAA I I = .

, , ( )0det 0A I = , . , 0 1= c 0c 0= .

:

A( ) ( ) ( )1 21 2( ) = "

( ) ( ) ( )1 21 2( ) = " , i i0 < . ( ) A n

(. ), . i 1 = ( ) ( )

7.4 A

, ( ) A( )( ) ( )1 2( ) = " , .

: ,

10 10 0 1

A =

.

, ( ) (2( ) 1 1 = + ) A

4 13

( )( ) 2( ) 1 1 1 = + = . ,

2

1 20 1 0 00 0 1

A I = =

= .

7.5 .

: 1A PBP= ( ), ( )A B .

1( ) ( )B BA P B P O = = ,

7.1, ( )A ( )B . , : 1B P AP=

1( ) ( )A AB P A P O = =

( )B ( )A . ( ) ( )A B = .

7.5 ,

.

. 1 1 00 2 00 0 1

A =

2 0 00 2 20 0 1

B =

,

( )( )( ) ( ) 1 2A B = = , ( ) (2( ) 1 2A ) =

( ) ( )2( ) 2 1B = . , A B .

7.6 . A TA

: ( ) ( ) ATA .

5 13

7.1 6.10, . 161.

* * *

7.2 ,

Cayley-Hamilton

, ,A B

A OM

O B = ,

AN

O B =

.

TT( ) ( )A A = = O [ ]TT( ) ( )A A O = = ( ) ( ) ( ) ( ) , ( ) ( ) = .

7.7 A O

EO B = E.K. .

. A B

: ( )E = E.K. . { }( ), ( ) ,A B ( ) ( ) p( )E A = ( ) ( )q( )E B =

, ( )E = ( )E = . , k( ) ( ) (k( ) ( )E ) < k( )E = , { }k( ) diag k( ), k( ) ,E A = B k( ) k( )A B= = 7.1 ( )A ( )B . , k( ) ( )E E.K. . ( )A . ( )B

6 13

: , ( )A ( )B .

A B M N

( ) ( ) ( )A B = , ( ) ( ) ( ) ( )det det det detI M I N I A I B = =

. ( ) ( )M N = =O* * *

7.3 1 13 1

A = , . 2005A

: ( ) ( ) 2det 4A I A = = , Cayley-Hamilton . , 2 4A = I ( )10022005 2 10024A A A= = A .

, .

( ) 1diag 2, 2A P P= .

( )( )( )

20042005 2004 2004 1

1002 1002 1 1002 1 1002

diag 2 , 2

diag 4 , 4 4 4 .

A A A P P A

P P A PP A A

= = = = =

1

* * *

7.4 1 0

1diag 0 1 , ,

0 00 0

J =

,

. ( )J : ,

. 7.7,

( )31( ) = ( 22 ( ) = ) )

)( 23 ( ) =

( )J = E.K. . { } ( ) (3 21 2 3( ), ( ), ( ) = .

* * *

7 13

7.5 3 3 A2 3 2A A I O + = .

: A ( ) 2p 3 2 = + ,

. :

( )(1= )2

I

( ) A ( ) 1 A I = = , ( ) 2 2A = = ( ) ( )p = , ( 7.4) ,

A

1A PDP=( )diag 1, 1, 2D = ( )diag 1, 2, 2D =

. P

, A ( ) ( )p = ,

A

( )diag , 2D I I = , + = .

* * *

7.6 1 0

0 10 0

A =

, 1 0

00 0

B

0 =

( )diag , , = ,

, . ( ) ( ) : .

,

( 3( ) ( ) ( )A B = = = ))( )3( )A = ( 2( )B = ( ) = .

, 7.5,

.

* * *

8 13

7.7 N

5 3 24 2 0

0 0 2A

= ,

1 00 10 0

B =

.

: y A

( ) ( ) ( ) (25 3 2 24 2A

I A = = = + )1 .

A ( )( )2A I A I O , . ( ) ( )A A

y ,

B2

1 0c cB B= + I2

2 2

2

2 10 20 0

B =

(1.3) 1 = 0, .

B

.

0c , c1

O

( ) ( )3B = * * *

7.8 , 2 4 3

0 0 01 5 2

A =

593 152A A A + = .

: ( ) ( )( )3 1 1A I A = = = + . ( ) 593 152 = + ( ) ( ) ( )0 1 1 0 = = = , ( )A ( ) . , Cayley-Hamilton

( ) 593 152A A A A O = + = .

* * *

9 13

7.9 1 2 20 3 40 2 3

A =

.

. ( ) . ( ) (3 102A I A I A I + )9I

9

. . 2005 20042 2A A A+ = + : . A

( ) ( ) ( )21 1A = + . ( ) ( ) ( ) ( )3 102 1 1 = +

( )A , ( ) ( ) ( )A = Cayley-Hamilton

( ) ( ) ( ) ( ) ( )3 10 92 AA I A I A I A A O + = = . . ( ) ( )( ) 21 1 = + = 1

I

, . A 2A =( ) ( )1002 10022005 2004 2 22 2A A A A A A+ = + = + 2I .

* * *

7.10 1 1 03 1 00 0 2

A =

.

. 4 3 26 2 7A A A A I + + . . ; A

. 1A 2A . A

: . ( ) ( )( )2 3 22 4 2 4A I A = = = + 8 ( )( ) ( )4 3 26 2 7 1 2 1A + + = + + .

Cayley-Hamilton 4 3 26 2 7 2A A A A I A I + + = .

. ,

( ) ( ) ( ) ( )2det 2 2A I A = = +

10 13

( ) ( )( )2 = + 2 ( ) ( ) ( )22 2 = + . ( ) ( )( )2 4A I= = +2 2 7.4, . A

,

,

2 =

( ) ( )d 2 3 rank 2 3 1 2A I= = = . . ( )0 8A = 0

O

,

. ( Cayley-

Hamilton)

( )3 1 2 31 8 det A = =A

3 22 4 8A A A I + = 1A

( )1 21 2 48A A A = I . 2A ,

( ) ( ) ( )2 1 21 1 1 12 4 2 2 4 88 8 2 16A A I A A I A A I A = = + = 2 I .

* * *

7.11 ( )p ( ) .

A ( )pB A=( ) ( )p

.

: ( ) ,

( )p ( ) ( )1 2q , q

( ) ( ) ( ) ( )1 2q p q + =1.

( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 2q p q p q det pA A A A I A A I A + = = 0 .

11 13

, ( ) { }1 2, , ,A = , ( )( ) ( ) ( ){ }1p p , , pA = . B ,

( )( ) ( ) ( ) ( ) ( )1 2 idet p p p p 0 p 0A = " i 1 . , 2, ,= ( ) ( )p , .

* * *

7.12 5 2 22 2 42 4 2

A =

.

. k\ kA I+ .

. .

. ( )1 1 318

A A = I , 2 1 112 108

A I = A I . ( )3 27 2A A= + : . A

( ) ( )( 23 6A I A = = + )I

.

, kA + { }k 3 k, 6 kA I+ = + + . k 3>. ( ) ( )( ) 23 6 3A = + = 18. , ( 7.4).

( )A A

, ,

6 =( ) ( )( )d 6 3 rank 6 3 1 2A I= = =

. A

. ( )0 108A = 0 ,

A

1A

( ) 2 3 18A A A A I O = =

(1 1 318

A A = )I . ( )2 11 1318 12 108A I A I = = 1 A .

12 13

Cayley-Hamilton 1A 2A

2 .

,

.

A

( )( ) 2 3 =