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 =