Μάθημα 8o La

20
Γραμμική Άλγεβρα ΙΙ Σελίδα 1 από 20 Μάθημα 8 ο ΚΑΝΟΝΙΚΗ ΜΟΡΦΗ JORDAN Έστω είναι ιδιοτιμή του πίνακα , αλγεβρικής πολλαπλότητας . 0 λ ν×ν A 0 1 ν > Ένα διάνυσμα , διάφορο του μηδέν, ονομάζεται γενικευμένο ιδιοδιάνυσμα, x τάξης , αντίστοιχο της ιδιοτιμής ρ 0 λ του πίνακα , ακριβώς όταν για τον A ελάχιστο φυσικό αριθμό είναι ρ , ( ) 0 A I x ρ −λ = 0 0 ( ) 1 0 A I x ρ− λ . (8.1) Για , τα γενικευμένα ιδιοδιανύσματα τάξης 1 είναι τα ιδιοδιανύσματα του 1 ρ= πίνακα. Προφανώς, ο αριθμός ρ είναι το πολύ ίσος με την αλγεβρική πολλαπλότητα του στο ελάχιστο πολυώνυμο 0 λ ( ) µ λ του . Τα διανύσματα A ( ) 1 0 x A I ρ− = −λ x , x , ( ) 2 2 0 , x A I x ρ− = −λ ( ) 1 1 0 x A I ρ− = −λ είναι γενικευμένα ιδιοδιανύσματα τάξης 1, 2, ,1 ρ ρ− αντίστοιχα του 0 λ , καθόσον ( ) ( ) ( ) ( ) k k k 0 k 0 0 0 A I x A I A I x A I x ρ− ρ 0 λ = −λ −λ = −λ = και ( ) ( ) k1 1 0 k 0 A I x A I x ρ −λ = −λ 0 . Έτσι από το γενικευμένο ιδιοδιάνυσμα x ρ x , τάξης ρ , δημιουργείται από την αναδρομική σχέση ( ) k 0 x A Ix k1 + = −λ , (8.2) ( ) 0 k 0,1, 2, , 1; = ρ− = x 0 , ένα σύνολο ρ γενικευμένων ιδιοδιανυσμάτων { } 1 2 1 , , , , x x x x ρρ = X . Το σύνολο ονομάζεται αλυσίδα γενικευμένων X ιδιοδιανυσμάτων παραγόμενη από το x ρ και το πλήθος ρ των διανυσμάτων αυτών ονομάζεται μήκος της αλυσίδας.

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Linear Algebra 8th

Transcript of Μάθημα 8o La

  • 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 = .

    * * *