Upload
scorober87
View
257
Download
29
Embed Size (px)
DESCRIPTION
Linear Algebra 8th
Citation preview
1 20
8
JORDAN
, . 0 A 0 1 > , , , x
, 0 , A , ( )0A I x = 0 0( ) 10A I x . (8.1) , 1 1 =. , 0 ( ) . A
( )1 0x A I = x , x, ( )22 0 ,x A I x = ( ) 11 0x A I = 1, 2, ,1 0 ,
( ) ( ) ( ) ( )k k k0 k 0 0 0A I x A I A I x A I x 0 = = =
( ) ( )k 1 10 k 0A I x A I x = 0 . x x , ,
( )k 0x A I xk 1+= , (8.2) ( )0k 0,1,2, , 1 ;= = x 0 ,
{ }1 2 1, , , ,x x x x = X . X x .
2 20
, 0 0 .
8.1 .
:
1 1 2 2c c cx x x 0+ + + =" . ( ) 10A I
( ) ( ) ( )1 11 0 1 1 0 1 0c c cA I x A I x A I x + + + =" 1 0 .
( )0 1A I x 0 = 20 2A I x, ( ) 0 = , , ( ) 10 1A I x 0 = , ( ) 10c A I x = 0 c 0 = . ( ) 20A I . , , 1c = 0
2 1c c 0 = = =" .
, (8.2) X
1 2 1 2
0 1 0 2 1 0 1
0
01 2
0
1 2
1
1
.
A x x x Ax Ax Ax
x x x x x
O
x x x
O
x x x J
= = + +
= =
% %
(8.3)
3 20
0
0
0
1
1
=
%%
O
JO
Jordan , . 0 . J
8.2 0 , , 0 ,
{ } ( )( ) (( )
0
0 0
# d
rank dim ker . = A I A I
= = )
: 0 . 1 ,
,
( )0d .
8.3 ( ) ( ) ( )1 2 k1 2 k( ) = " A ( ) jj jkerV A I = ,
n1 2V V V= ^ k
. jdim V j
.
4 20
8.4 ( ) ( ) ( )1 2 k1 2 k( ) = " , A j
( ) ( )j j 1j jker kerA I A I + = .
j j . j 1 + .
8.5 , 0 A
( ) ). , ( )( ) (kk 0m dim ker , k 1,2, ,A I= = k k k 1s m m = . k :
, ( ) (k 1 k0ker kerA I A I )0 0 . ks k
( ) ( )( )k k0 0rank dim kerA I A I = , : 0 A ( ) ( )k 1 kk 0s rank rankA I A I= 0 (8.4) , k ( )k 1,2, ,= , . 0 ( )
k ks s 1+ , k
k 1+ ,
5 20
. Jordan k
A k ks s 1+ Jordan 0 , k k .
8.6 , 0 A 0 0 .
.
0 , ( ) A : 0
( )0A I x = 0 , ( ) 10A I x 0 1 2 s, , , x x x , . ( )0 j j 1,2, ,s = A I x ,
. 1 ( ) 10A I y = 0 , ( ) 20A I y 0
1 1s s =
11 2, , ,y y y 1 ,
( ){ }j 0 j, ,x A I x . j 1, 2, ,s=
( ) ( ){ }( ) ( )
10 1 0 s 1 2
1 20 0
span , , , , , ,
ker ker .
=
A I x A I x y y y A I A I\
( ) ( )20 j j 1,2, ,s = A I x , ( ) ( )0 u 1u 1, 2, ,A I y =
6 20
2 , ( ) 20A I 0 = , ( ) 20A I 0
2 2s s 1 =
21 2, , , 2 ,
. ,
( ) ( ) ( ) ( ){ }( ) ( )
1 2
2 2 20 1 0 s 0 1 0 1
2 30 0
span , , , , , , , ,
ker ker .
=
A I x A I x A I y A I y A I A I\
. 0
, M
(8.3),
1 2 k, , , , :
1 211 1 21 2 k1 k kM M M M M M M = " " " "
i1 i iM M " i .
( )
1
1 1
1
11 1 k1 k
11 11 1 1 k1 k1 k k
11 1 k1 kdiag
k
k k
k
AM AM AM AM AM
M J M J M J M J
M J J J J
=
=
=
" " "
" " "
(8.5) Jordan i1 i, , iJ J i . M , Jordan . A
(8.5)
( )111 1 k1 kdiag kJ J J J J = Jordan . A
7 20
8.1
Jordan
1 1 0 10 1 0 00 0 1 10 0 0 1
A
= .
: A ( ) ( )41A = . (8.4) ( ) ( )21 =
( ) ( )220 1 0 10 0 0 0
s rank rank rank 0 20 0 0 10 0 0 0
A I A I
= = =
=
( ) ( ) ( )0 11 4s rank rank rank rank 4 2 2A I A I I A I= = = .
:
[ ] [ ]3 1 0
3 1 2 1diag 0 3 1 , , , 0 , 0
0 3 0 20 0 3
= J
Jordan 9 9 . : A ( ) { }0, 2, 3 =A . . / ( )1 3 5 = , . / ( )2 2 2 = , ./ ( )3 0 = 2 ,
( ) ( ) ( )2 52 2 3 = A . , , . ( )d 3 2= ( )d 2 1= ( )d 0 2= . ( ) ( ) ( )3 23 2 =
8 20
, 1 = ,
, .
2x 2x 2 XX 2 ( )2A I x 0 =
x
[ ]T2 0 1 0 0x = , [ ]T2 0 0 0 1x = , (8.1), ( ) 2A I x 0 ( ) 2A I x 0 . (8.2)
1x 1x
( ) [ ]T1 2 1 0 0 0x A I x= = , ( ) [ ]T1 2 1 0 1 0x A I x= = .
{ }2 1,x x= X , { }2 1,x x= X . [ ]1 2 1 2M x x x x= #
1 1 1 1diag ,
0 1 0 1AM M
= .
Jordan . A
* * *
8.2 Jordan
2 0 0 10 2 0 11 1 2 0
0 0 0 2
A
= .
: ( ) ( )42A I A = = ,
( ) ( )32 =
0 0 0 10 0 0 1
21 1 0 0
0 0 0 0
A I
= , ( )2
0 0 0 00 0 0 0
20 0 0 20 0 0 0
A I
=
, ( )rank 2 2A I = ( )2rank 2 1A I = .
9 20
,
3 .
( ) ( )2 33s rank 2 rank 2 1 0A I A I= = 1=
0( )32A I x = , ( ) [ ]T22 0A I x 0 x = 0 0 1 . 3
( ) ( ){ }[ ] [ ] [{ }
2
T T
2 , 2 ,
0 0 2 0 , 1 1 0 0 , 0 0 0 1 .
=
A I x A I x x
=
X
]T1=
( ) ( )22s rank 2 rank 2 2 1A I A I= = 2 2 3s s 1 1 0 = = = , ,
.
2
( 2A I )x=
X
( ) ( )01s rank 2 rank 2 4 2 2A I A I= = 1 1 2s s 2 1 1 = = = , .
( ) [ ] [ ]T T1 22 1 1 0 0 , 0 0 1 0A I x 0 = = = . ,
. ,
2 ( )22A I xX
0 1 0 10 1 0 12 0 0 00 0 1 0
=
####
M ,
2 1 0 00 2 1 00 0 2 0
0 0 0 2
J
=
###
" " " # "#
1A MJM= .
* * *
8.3
2 0 1 13 5 4 14 3 3 11 0 1 2
= A .
10 20
: ( ) ( ) (2 21 2A I A = = ) ( ) ( ) = A . ,
1 =
1 0 1 13 4 4 14 3 4 11 0 1 1
= A I , ( )2
2 3 2 10 4 4 42 0 1 42 3 2 1
= A I .
[ ]T2 2 1 0 1x =
2
( ) [ ]T1 2 3 3 6 3x A I x= = . 2
, 2 =0 0 1 13 3 4 1
24 3 5 1
1 0 1 0
A I
= , ( )2
3 3 4 16 3 4 2
210 6 8 2
4 3 4 0
A I
=
T
23 31 04x
= 4
2 . , ( ) T1 2 3 3 32 04 4 4x A I x = = .
[ ]1 2 1 2M x x x x= # , 1 1 2 1diag ,0 1 0 2J =
. 1A MJM=
* * *
8.4
2 2 11 1 11 2 2
A =
.
: ( ) ( 31A I A = = ) ( ) ( )21 = . 1 =
11 20
1 2 11 2 11 2 1
A I =
2
( )2A I x 0 = , ( )A I x 0 . [ ]T2x = , 2 + [ ]T2 0 1 1x = .
( ) [ ]T1 2 1 1 1x A I x= = . { }1 2,x x Jordan
2
1 10 1 .
2 ,
( ) ( )22s rank rank 1 0A I A I= = 1==
]2
.
,
. ,
( ) ( )0 11s rank rank 3 1 2A I A I= = A1x
( ) [ ] [1 T T2 1 21 2
cc c 1 0 1 c 0 1
c 2cA I x 0 x
= = = + + .
[ ]T1 0 1 [ ]T0 1 2
1x
{ }1 2,x x (8.5).
M
1 0 11 1 11 1 2
M =
, . 1 0 11 1 01 1 1
M =
AM11 1 00 1 0
0 0 1
= =
##
" " # "#
J M
* * *
12 20
8.5
0 4 23 8 34 8 2
= A .
: ( ) ( )32A I A = = , ( ) ( )22 = 2 4 2
2 3 6 34 8 4
= A I .
( ) ( )22s rank 2 rank 2 1 0A I A I= = 1=
2=
0=
( ) ( )0 11s rank 2 rank 2 3 1A I A I= = ( ) .
A 2 1 1 2s s 1 = =
, ( )22A I x ( )2A I x 0 (*) x
[ ]T1 2 3x x xx = 1 2x 2 x x3 + . [ ]T2 0 0 1u = , (*)
A 2
( ) [ ]T1 22 2 3u A I u= = 4 , { }2 1,u u=X .
( )2A I x = 0
2 2
A
1 2 3
2 2 1 2 1 1
3 3
x 2 x x 2 1x x c 1 c 0 c cx x 0 1
x + = = = + = +
.
1u 1 2,
[ ]1 2 1M u u = # [ ]1 2 2M u u = #
13 20
[ ]1 2 1diag , 20 2
M AM =
.
* * *
8.6
, Jordan .
( 32 ) 3 3A A
: ( ) A( ) ( )31 2 = ( ) ( )22 2 = ( )3 2 = .
12 1 00 2 10 0 2
J =
,
[ ]2 2 1diag , 20 2J =
( )3 diag 2, 2, 2J = .
* * *
8.7 5 5 , . A ( ) ( )32 = : .
Jordan , Jordan 3
A ( ) ( )52A = A 3 ,
1
2 1 00 2 10 0 2
J =
1
2 1diag ,
0 2J J
=
( ) ( )22s rank 2 rank 2 2A I A I= = (. 2 ),
14 20
( ) { }5 rank 2 2 # = = A I , ( )rank 2 3A I = ( )2rank 2 1 =A I .
[ ] [ ]( )1diag , 2 , 2J J=
( ) ( )0 11s rank 2 rank 2A I A I= 3= .
* * *
8.8 , A x ^ , , ( )A , 1A ,
. ( )1 1A : x
( )A I x = 0 0, ( ) 1A I x .
(*) ( ) ( ) ( )1 1A I x A I A x = = 0
0
(*) ( ) ( ) ( )1 1 11 1A I x A I A x = x 1A , . , (*) ,
,
.
A
x A
* * *
15 20
8.9 , . A : , ,
A 0 =
0x 0
Ax x A x x x 0 = = = = .
, ( ) = A ( ) = , . Jordan A
( )1diag , ,J J J=
i
0 1
1
0
=
% %%
i i
O
JO
. 1 + + = " iJ i{ }1max , , = ,
1A MJ M O = = , J O . =
* * *
8.10 Jordan
0 1 00 0 10 0 0
A =
.
: , 3 3 X 2X A= . , Jordan ,
1X MJM= JX
( ) ( ) { } ( ) { }2 1 2 0 0MJ M A J A J = = = = .
,
16 20
J A= [ ]0 1diag , 00 0
J = ( )diag 0, 0, 0J = .
20 0 1 0 1 00 0 0 0 0 10 0 0 0 0 0
J A MJ AM M M = = =
.
3 M
, , . M
, J2MJ AM AM O A O= = = ,
.
, J O X O A O= = = , .
* * *
8.11
2 31 0 10 1 1
A =
Jordan.
: 6.9
,
A
0 = 1 = ( )d 1 1 2= . , 2 = 1 = ( )d 1 1 2 = .
, . 0 = ( ) ( )( )21 1A = + ( ) (A A ) , , , . , ( ) ( )( )1 1 = + A A
1 0 31 1 10 1 2
A I =
, ( )21 3 30 0 01 3 3
A I =
17 20
( )2 1 2 3 1 2 1 2 23 3
x 3x 3x 0 c 1 c 0 c c0 1
= + = = + = + A I x 0 x u v2 .
,
( ) ( ) . 2 2,u v A 2
2 A I u 0 2 A I v 0( ) [ ]T1 1 3 2 1= =u A I u
{ }2 1,u u . 1 = ,
( ) [ ]T3 0 31 1 1 c 1 0 10 1 0
A I x 0 x 0 x + = = =
.
, , 1A MJM=1 3 30 2 11 1 0
M =
###
, [ ] 1 1diag 1 ,0 1
J =
.
, 2 = ( ) ( ) ( )21A = + 1 ( ) (A A ) . ,
1 2 11 1 10 1 0
A I + =
, ( )23 3 12 2 01 1 1
A I + =
( ) [ ]T2 c 1 1 0A I 0 + = = , c . [ ]T1 1 0 = ,
2
( )A I 0+ ( ) [ ]T1 0 1 A I = + = { }, . , 1 =
( ) [ ]T1 2 1
1 1 1 3 2 10 1 2
A I x 0 x 0 x = = =
.
,
18 20
3 12 01 1
110
=
###
M , [ ] 1 1diag 1 ,0 1
J =
.
. 1A MJM=
* * *
8.12 Jordan
3I BAO B = ,
T13
B = [ ]T1 1 1 = . : , ,
B 1 =
( )T T1 13 3B = = = . , rank 1 3B = < B 0 = , ( )d 0 3 rank 2B= = . ,
B 3
0 = . ( ) ( )2 1B = ( ) ( )42 1A = .
( )2 T T T1 19 3B B= = =
( )2
26
O B O OA I
O B I O B I = =
( )2 36 I B O OA A I OO B O B I = = .
, A ( ) ( )21A = . ,
1 =2
19 20
( ),
2
( )2 1 1
1rank rank 1 2 1 23
1 1 2B I
= = ,
( ) ( )22 6 6s rank rank 3 2A I A I= = 1=
=
( ) ( )01 6 6s rank rank 6 3 3A I A I= = . ( )26A I x 0 =
12 2 1 2 1
2
, ,xO O
0 Bx x x x xxO B I = = = .
, 0
x = ( )6
B x A I x
0 0 0= = =
,
,x x 2 1 = . ( )6A I 0 =
( )1 2 2 1 22
, ,O B
0 B B I 0 0 O B I = = = = 1 .
[ ]T1 0 0 0 0 0# [ ]T0 1 0 0 0 0# . ,
.
0 = rank 4A =( 6 rank 2A = ) Az 0=
1 1 2 1
2 2
z z Bz 0 zI B0
z Bz 0 Bz 0O B+ = = = = = 2
0
2
1 2 1 2 11 1 1 1 1
, 1 1 1 , c 1 c 01 1 1 0 1
z 0 z 0 z 0 z = = = = +
.
,
20 20
1 0 1 0 0 01 0 0 1 0 01 0 0 0 00 1 0 0 1 10 1 0 0 1 00 1 0 0 0 1
= 0
# # # ## # # ## # # ## # # ## # # ## # # #
M ,
[ ] [ ] [ ] [ ]1 1 , 1 , 1 , 0 , 00 1
= J diag
. 1A MJM=
* * *
1. Jordan
5 3 14 2 14 3 0
A =
, 3 1 0
4 1 34 2 4
B =
,
3 6 3 22 3 2 21 3 0 11 1 2 0
= .
2. Jordan
, . 5 5A ( ) ( )21A = , 7 7B ( ) ( )( )52 2 1B = , ( ) ( )( )22 2 1B = .
* * *