Μάθημα 7ο Γραμμική
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Transcript of Μάθημα 7ο Γραμμική
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1 10
7
AA A A =
: : (5.24)
, . 125,
6.12, . 145,
6.16 ( Schur), . 154
7.1 ( ) A .
: A PDP= , P D. D D
AA PDD P PD DP A A = = = . , . A A
A PTP= , T, ( Schur). AA A A =
TT T T = . T
T ( 7.1).
, . A kA k` , B A
2B A= .
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2 10
( )2AA A A A = = 6.9 (. 144) :
A
.
7.2 A
( )gA A = , g . : A A PDP= ,
( )1 2diag , , ,D = [ ]1 2P x x x= PP I = .
[ ] ( )
[ ] ( ) ( )
1
21 2 1 2
1
21 1 2 2 1 1 1
1 1 2 2
diag , , ,
(7.1)
xx
A x x x
x
xx
x x x x x x x
x A A A ,
= = = + +
= + + +
#
"#
" . (7.1) i iA x x= i iA
1 1 2 2A A A A
= + + + " .
( ) ( ) ( ) ( )1 1 2 2g f f f = + + + " , Lagrange ( )if
( ) ( ) ( )( ) ( )( ) ( )( ) ( )1 i 1 i 1i i 1 i i 1 i i 1 if +
+
= " "" "
, , i( )i if 1 = ( )i jf 1 = j ( )ig i = .
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3 10
( ) ( ) ( ) ( ) ( )( )( )
i i i 1 i i i
i i i
f f diag f , , f , , f
diag 0, , 1, , 0
A P D P P P
P P x x A
= = = = =
(7.1)
( ) ( ) ( ) ( )1 1 2 2 1 1 2 2g g g gA A A A A A A = + + + = + + + =" " A . , , , ( )gA A = g
( ) ( )g gAA A A A A A A = = = , , . A
(7.1) ,
A
(7.2) 1 1 2 2 k kA A A A= + + + " i iA X Xi= iX i A i .
7.3 , A B
.
: ( )A 1 2 r, , ,x x x . AB BA=
( )i iABx BAx Bx= = i , . iBx A
(7.3)r
i ij jj 1
; i 1, 2, , rBx p x=
= = (7.3) BX XP= , [ ]1 2 rX x x x=
r r ij .P p = , , P X , ,A B ( ) ( )B X XP X= = AX X= ( ) ( )A X X= .
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4 10
7.4 , AB BA= Q
1A QTQ= , 2B QT Q= ,
. 1 2,T T
: , 1 = , [ ]1Q = . , S
( ) ( )1 1 . , A B AB BA= , A B( 7.3) , BxAx x= x= ,
21x = .
[ ]R x X= # , R R I = , ,
x AXR AR0 x AX
=
x BXR BR0 x BX
=
.
, A B
R A , R R BR
( ) ( )1 1 . 1A X AX= 1B X BX= ( ) ( )1 1 1A 1BS ,
1S A S T
1= , 1 2S B S T =
, . , 1T 2T1 0
Q R0 S = ,
Q Q I =
1
11
1 1 1
,
0 0 0 x AXQ AQ R AR
0 S 0 S 0 S 0 S0 A
x AX T
0 T
= = = =
1 0
22
1 10 0 x BXQ BQ R BR T
0 S 0 S 0 T
= =
= .
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5 10
7.1 ,
.
:
11 12 1
22 2
TO
=
% #
. TT T T = , (1.1)
22 2 211 12 1 11 12 1 0 + + + = = = =" " .
, (2.2) 2 22 2
22 23 2 22 23 2 0 + + + = = = =" " . , ij 0 = i j ,
( )11 22diag , , ,T = .
* * *
, AB BA= A B Q
1Q AQ D = , , 2Q BQ D =
. 1D 2D
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7.2 1 2, , , , ij i, j=1A =
22ii 1 i, j 1
= = ij ( Schur ).
. A
: Schur ,
A PTP= A P
ij i, j=1tT
=
. ii it = ( )AA P TT P = ( ) ( )( ) ( )tr tr trAA TT P P TT = = .
( ) 2iji, j 1
tr AA=
=
( ) 2 22ij i iji, j 1 i 1 i, j 1
i j
tr t tTT = = =
= = + .
22i ij ij
i 1 i, j 1t 0, i j
= = = = T . A
* * *
7.3 . , k A `. , kA O A O= =. . kA I A= :
. A
( )1 2diag , , ,A P P=
. PP I = kA O= , kj 0 = .
j 1, 2, ,= A O=
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. , k kj 1A I= =2 ik
j jj 1, 2, , 1 e = = = ,
{ }0,1, , k 1 . , 1 2
2 i 2 i 2 ik k kdiag , , ,A P e e e P =
.
* * *
7.4 iA B = + ^ , .
,B \
. , B B,B = .
. . BKB=
: ( )( ) ( )( )T T T Ti i iAA A A B B B B = + = + i T T T TBB B B+ = + T T T TB B B B = . . , TB B= B . T= B
B +
= . ,
T T T T T TT
T T T T T T
T T T TT
T T T T .
B B BB BKKB B B B BB
B B B B K KB B B B
+ = = + = = + +
* * *
7.5 5 72 4
A =
,
P
T TA PTP= .
:
Schur. A ( ) { }2, 3A = , 2 =
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[ T1 1 12
x = ] .
{ } [ ]T1span span 1 12
x =
1 111 12
P =
.
2 90 3
TP AP T = = .
* * *
7.6 , A
. ker kerA A=. ( )Im kerA A =III. Im ImA A= : . . kerx A A
0 ker ker kerA x Ax A x 0 x A A A = = = .
, A ( )ker ker kerA A = A . ker kerA A=. ( )ker kerx A = A
ker
( ) ( ) ( ) ( ), 0 , ker Imy A x y x Ay y Ay A A A = = ^ D D . , ( )Imx A
y ^ , ( ) ( )0 , 0 ker kex Ay y A x y A x 0 x A A = = = =D D rA
,
. ( ) ( )Im ker ker ImA A A . ( )Im kerA A =. , .
* * *
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7.7
Im ,
A B Im A
B A B+ . : , Im ImA B ,x y ^ ( . III ( 7.6)
) ( ) ( ) ( ) 0 0Ax By By Ax y B Ax x A By B A A B O = = = = =D D =BIm ImA
, .
( ) ( ) ( ) ( ) 0A x B y B y A x = =D D BA AB O = =
( )( ) ( )( )( ) ( ).
A B A B A B A B AA BB AB BA AA BB
A A B B B A A B A B A B
+ + = + + = + + + = += + + + = + +
* * * 7.8 . A
. A , A . ( ){ }2 max :A A= . : . , A 1 = ^ ^ " ^ { }i span x=^ i . di
ix
A m k= < , . 1= ^ " ^ k , 6.12, . 147,
( ) ( ) ( ) ( )( ) ( ) ( )
1 1 2 2 k k 1 1 2 2 k k
1 1 1 2 2 2 k k k
c c c c c c
c c c .
A A x x x A x A x A x
x x x
= + + + = + + += + + +
" ""
. , Ax x= A x x = 2A Ax x = .
( ){ } ( ){ }2 max : max :A A A= = A .
* * *
7.9
, .
A ,A B
,A B
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: AB BA= 2 2A B AAB ABA BAA BA= = = =
kA B BAk= , k` . , ,
A
( )g ( )gA = A ( 7.2). , ( ) ( )g gA B B A=
( ) ( )g gA B A B B A BA = = = .
* * *
7.10 A ,
, , .
1 2 q, , ,x x x
A XX= 1 2 qX x x x= A : ,
,
1 2 k, , , Ai =
( )11 q k qdiag , , kA P I I P=
1 1 k kA A A A= + + + + " " .
( ) ( ) ( )( ) ( )( ) ( )( ) ( )1 i 1 i 11 i 1 i 1s s s s
f s + +
= " "" "
k
k
,
( )f =1 ( )jf 0 = , j 1, 2, , k ; j i= .
( ) ( )( ) ( )( )( )
11 q q k q q
q
f f diag , , , , diag 0, , f , , 0
diag 0, , , , 0 .k
A P I I I P P I P
P I P XX A
= = = = =
* * *