Μάθημα 9ο Γραμμική
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Transcript of Μάθημα 9ο Γραμμική
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1 6
9
KRONECKER
9.1 ,A B , 2( ) + . : , :
2 2 2
2 2
( ) ( ) ( ) 2( )( )
2 =
+ = + + + +
* * *
9.2 , : ,A B
. de t( ) (det ) (det ) =B I. , . 1 1( ) = B 1
)
)
,A B
II. rank( ) (rank )(rank ) = V. ( )(tr( ) tr tr = : . VI 1.1 (. 16) :
( ) ( )( = B
det( ) det( ) det( ) = B I .
det( ) det[ diag( , , , ) ] (det )B B B B = =
: : 6, . 15.
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2 6
T 2det( ) det[ ( ) ] (det ) det( ) (det ) = = = .
I. 1 1 1 1( )( ) ( ) ( ) = = =B B
, : 1 1( ) 1 = B B .
II. rank = rank = . 3.1 (. 48), , ,P Q ,S T
I OPAQ
O O = ,
= I
SBTO
.
( )( )( ) ( ) ( ) = = = I I I
P S A B Q T PAQ SBTO O O
,
, rank( ) (rank )(rank ) = = . IV.
( ) ( )( )( ) ( )( )
11 11 11 22 11 22 11
11 11 11
11 11
tr( )
tr tr
= + + + + + + + + + = + + + + + += + + + + =
A B
* * *
9.3 0 ii 0
= .
: A
2
i idet( ) det det( ) ( 1) ( 1) ( 1)
i i = = = = +
I II I
I I .
, ( ) A { }( ) 1, 1 = . A .
* * *
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3 6
9.4 ( , ) x ( , ) y , :
,A
. ( )( ) ( ) = x y x y . ( ) ( ) + = + x y ( ) : A B = Ax x .
= By y
. ( ) , ( ) ( = = = x y Ax By x y x y ) x y . . ,
,
.
( )( ) ( )( ) + = + = + = + x y Ax y x By x y x y x y
+ * * *
, , A Ae :
Ae2
1! 2! !
= + + + + +
A A AI
A B A Be e+ = e =AB BA .
9.5 A I I B A Be e + = e : :
0 0 0
( )! ! !
A I AA I A I Ae I
= = =
= = = = e I)
.
, ( )( ) ( )( ) ( = = ( )( )A I I B A I I B A B A Be e e e e e e + = = =
* * *
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4 6
9.6 ( ) , Adet( ) ( 1) det[ ]I A ( A + = ) .
: 1 2( ) | | ( )( ) ( ) = = ,
1 2( ) ( 1) ( )( ) ( ) = + + +
1
( ) ( 1) (
=
) = + . (= ) )( , ( ) , (, . 135)
2
1 11 1
( 1)
, 1 , 1
( 1) det ( ) ( 1) ( ) ( ) ( 1) ( ) ( )
( 1) ( ) ( ) det( )
+
= = +
= =
= = + +
= + = + = +
A I I A
, 9.4, II, , ( ) + + .
( 1)( 1) 1 + = , , 1 + .
* * *
+ =
,A B ,X
.
+ = (9.1) Liapunov
.
1.2, . 17, (9.1)
=Gx (9.2) , = + G A I I B ( )x = ( ) = .
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(9.2)
:
I. ( 3.11, . 68) rank( , rank=G ) G II. ( 4.18, . 102) (ker ) * G , det 0G , ( 9.6) : ,A B
0 + . (9.1) ,
+ = (9.3) Liapunov.
, (9.1) 0 (9.3),
0 0( ) ( ) + + + = , (9.1) 0 = + . 0 + .
9.1 (9.1)
= ,
=
.
: (9.1) . X
I X
I=
A O I X A AX A XBM N
O B O I O B O B + = = =
= ,
, , 1M N = det 1= . , . ,M N 1M RNR= M , N R
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1 2
3
= R R
RO R
,
, . 1R 3R
=MR RN 1 1=AR R A 2 2 1+ =AR R B R .
1 1 1 1 11 2 1 2 1 2 1 2 1 2 2( ) + = + = +AR R R R B R AR R R B R AR R B = ,
, (9.1) 11R R2
.