ku2 (1108608), страница 2
Текст из файла (страница 2)
(+> ,&2->-)+$:/)4-.) A 2 Mn n 5 . , ! A 2 Mn n 5 ... x2. ""#$% &'() #*+x802. $@ *-)'(0< A 2 Mn &=&>1-3(*Ri (A) =0nXj =1j =ijaij j6(A) = f1 : : : ng { *1&A+/)4& /&=/)4+11<6 >1-3+1(8.% 1 (%8*%). A 2 Mn -":(A) nfz 2 Cn : jz ; aiij Ri(A)g :(1)i=1$ , % k , 1 k n & , ' n ; k , 0 k A .:"#&. 9./): { /&=/)4+11&+ >1-3+1(+ *-)'(0< A ( x =(x1 : : : xn) 6= 0 { /&&)4+)/)4.LC(8 /&=/)4+11<8 4+2)&': Ax = x .
=&>1-3(*jxpj = maxi=1:::n jxij 6= 0. -2 2-2 Ax = x , )&xp = (x)p = (Ax)p =(xp( ; app) =nXj =1j =pnXj =1apj xjapj xj6$+,&4-)+$:1&,jxpjj ; appj X japj jjxj j jxpj X japj jnnj =1j =pj =1j =p6(6nXj ; appj japj j:j =1j =p6-2(* &='->&*, /&=/)4+11&+ >1-3+1(+ 2 (A) 7'(1-,$+A() &=S+,(1+1(L 2'.%&4 4 (1).&2-A+* 4)&'&+ .)4+'A,+1(+ )+&'+*<. 9&$&A(* A = D + B , %,+ D =diag(a11 : : : ann) 2 Mn , B = A ; D T A" = D + "B , " > 0. &%,- Ri (A") =Ri("B ) = "Ri(A). +> &%'-1(3+1(@ &=C1&/)( *< *&A+* /3()-): 3)& 7+'4<+ k000..
x2. ""#$% &'() #*+812'.%&4 4 (1) &='->.L) /4@>1.L &=$-/):, 1+ 7+'+/+2-LC.L/@ / &/)-$:1<*( n ; k2'.%-*(. =&>1-3(*kfz 2 Cn : jz ; aii j Ri (A)g i=1nGn k (A) =fz 2 Cn : jz ; aiij Ri(A)g Gk (A) =00;i=k+17'(3+* 7& ./$&4(L\Gk (A) Gn k (A) = :-//*&)'(* 44+,+11<+ *1&A+/)4- ,$@ *-)'(0< A" :kGk (A") = fz 2 Cn : jz ; aiij "Ri(A)g i=1nGn k (A") =fz 2 Cn : jz ; aiij "Ri(A)g :(2);00;&%,- ,$@ 4/+6 " 2 U0 1]i=k+1Gk (A") Gk (A1 ) = Gk (A) Gn k (A") Gn k (A1 ) = Gn k (A):;;(3);7'(3+*T*1&A+/)4& Gk (A1 ) = Gk (A) /4@>1& 7& ./$&4(L. /($. (2), (3)Gk (A") Gn k (A") = ,$@ 4/+6 " 2 U0 1]. =&>1-3(* 3+'+> i(A") i -+ /&=/)4+11&+ >1-3+1(+ *-)'(0< A" , (A") = f1(A") : : : n(A")g { /7+2)' *-)'(0< A" .9& 7&/)'&+1(L i(A1 ) = i(A) = i , i(A0) = aii .&=/)4+11&+ >1-3+1(+ i(A") *-)'(0< A" @4$@+)/@ 2&'1+* 6-'-2)+'(/)(3+/2&%& *1&%&3$+1- ;)&8 *-)'(0<. &'1( *1&%&3$+1- @4$@L)/@ 1+7'+'<41<*(5.120(@*( +%& 2&;55(0(+1)&4.
$+,&4-)+$:1&, /&=/)4+11<+ >1-3+1(@ i(A")*-)'(0< A" @4$@L)/@ 1+7'+'<41<*( 5.120(@*( ;$+*+1)&4 ;)&8 *-)'(0<, 2&)&'<+, 4 /4&L &3+'+,:, @4$@L)/@ 1+7'+'<41<*( 5.10(@*( 7-'-*+)'- " . -2(*&='->&*, 2-2 2&*7&>(0(( 1+7'+'<41<6 5.120(8, /&=/)4+11<+ >1-3+1(@ i(A")*-)'(0< A" @4$@L)/@ 1+7'+'<41<*( 5.10(@*( 7-'-*+)'- " .9& ,&2->-11&*. 7+'4&*. .)4+'A,+1(L )+&'+*< (A") Gk (A") S Gn k (A"),$@ 4/+6 " 2 U0 1], ).+.;;i(A") 2 Gk (A") Gn k (A"):(4);9& ,&2->-11&*.
4<K+\Gk (A") Gn k (A") = (5),$@ 4/+6 " 2 U0 1], 7'(3+* i(A1) = i(A) = i , i(A0) = aii . $+,&4-)+$:1&,;i(A0 ) = aii 2 Gk (A0) ,$@ 1 i k i(A0 ) = aii 2 Gn k (A0) ,$@ k + 1 i n:(6);.. x3. - . ) &'() #*+829&/2&$:2. i(A") { 1+7'+'<41-@ 5.120(@ ( 7'(1(*-+) 4/+ 7'&*+A.)&31<+ >1-3+1(@, )& (> (4), (5), (6) 4<)+2-+), 3)&i(A") 2 Gk (A") ,$@ 1 i k i(A") = aii 2 Gn k (A") ,$@ k + 1 i n,$@ 4/+6 " 2 U0 1]. 9'( " = 1 *< 7&$.3-+* 4)&'&+ .)4+A,+1(+ )+&'+*<.;x3. 9./): 1-6&,@)/@ /&=/)4+11<+ >1-3+1(@ *-)'(0< A 2 Mn 9./): W{ -$%&'()*1-6&A,+1(@ /&=/)4+11<6 >1-3+1(8, b (A) = W(A) { 7&$.3+11<8 ;)(* -$%&'()*&* /7+2)' *-)'(0< A .
F)&) /7+2)' 1+ /&47-,-+) / (/)(11<* >1-3+1(+* (A)/7+2)'- *-)'(0< A , 7&/2&$:2. 4 /($. )+&'+*< 1.1 1-8)( 4/+ /&=/)4+11<+ >1-3+1(@ *-)'(0< A >- 2&1+31&+ 3(/$& -'(5*+)(3+/2(6 &7+'-0(8 1+4&>*&A1&.=&>1-3(* 3+'+> A + E *-)'(0., /7+2)'&* 2&)&'&8 @4$@+)/@ 1-=&' b (A), ).+.b (A) = (A + E ). 0+1(* 7&%'+K1&/): 7'( 1-6&A,+1(( /&=/)4+11<6 >1-3+1(8().+. b (A) ; (A)) 3+'+> 4+$(3(1.
*-)'(0< E . 1. ( D = diag(1 : : : n) { , E = (eij ) 2Mn . * b 2 (D + E ) { D + E , = i 2 (D) { D , , jb ; j kEk(kk11{ , . . 9).:"#&. 9'(*+1@@ )+&'+*. !+'K%&'(1- 2 *-)'(0+ D +E , 1-6&,(*,3)& /.C+/)4.+) )-2&+ i , 1 i n , 3)&89>>>>n<Xb 2 z 2 Cn : jz ; i ; eiij Ri (E ) = jeij j= :>>>>j =1:0j =i6-2 2-2 jz ; i ; eiij jz ; ij ; jeiij , )&89n<=Xb 2 :z 2 Cn : jz ; ij jeij j :j =1&nXXjeij j i=1maxje j = kEk:::n j =1 ijj =1n19&;)&*.b 2 fz 2 Cn : jz ; ij kE k3)& ,&2-><4-+) .)4+'A,+1(+ $+**<.1..:g x4.
&.+ &83% 1. ( A 2 Mn { , .. ' C X = diag(1 : : : n) , A = C XC 1 . ( E = (eij ) 2 Mn . * b 2 (A +E ) { A + E , = i 2 (A) { A , , jb ; j (C )kEk;11 (C ) { C (. . 13).1:"#&. -)'(0< A + E ( C 1 (A + E )C = X + C 1 EC 7&,&=1< (;;7&)&*. (*+L) &,(1-2&4<+ /&=/)4+11<+ >1-3+1(@. /($. $+**< 1 ,$@ 4/@2&%&b 2 (A + E ) = (X + C 1EC ) /.C+/)4.+) = i 2 (X) = (A) )-2&+, 3)&;jb ; j kC1 EC;k kC 1k kEk kC k;1111= (C )kE k :11+&'+*- ,&2->-1-.x4. )+7+11&8 *+)&, 7&>4&$@+) 1-6&,(): *-2/(*-$:1&+ 7& *&,.$L /&=/)4+11&+>1-3+1(+ ( /&&)4+)/)4.LC(8 +*.
/&=/)4+11<8 4+2)&' ,(-%&1-$(>('.+*&8 *-)'(0< A 2 Mn .x 4.1. 5 "*%% 1 (5 $). ( A 2 Mn ' ei , i = 1 : : : n : Aei = i ei ,(ei ej ) = ij , j1j > j2j j3j : : : jn (.. , { ). * x(0) 2 Cn , (x(0) e1) 6= 0 , x(k+1) = Ax(k) (k+1) (k)(1k) = (x(x(k) x x(k) ) ) k = 0 1 : : :(1) ' 1 ( ' ), 0 k 1(1k) = 1 + O @1 2 A (k ! 1)1.. x4. &.+ &84 e(1k) = kxx(k) k , ' 1 ( ' ):0 k 1(k)e1 = ei'e1 + O @ 2 A 1(k) ei' { , ' 1 .:"#&. 9&/2&$:2. 4+2)&'- e1 : : : en &='->.L) =->(/ 4 C n , )&nx(0) = P ciei , 7'(3+* 7& ./$&4(L c1 = (x(0) e1) 6= 0.
<3(/$(*i=1X! XnnnX(k)k(0)kx =A x =Aciei = ciAk ei = ki ei:i=1i=1i=19&;)&*. x(k) = c1k1 e1 + O jk2 j , x(k+1) = c1k1+1e1 + O jk2+1j . -$++, 4<3(/$(*kx(k)k = (x(k) x(k)) = c1k1 e1 + O jk2 j c1k1 e1 + O jk2 j= jc1 j2j1j2k + O jk1 jjk2 j T (x(k+1) x(k)) = c1k1+1e1 + O jk2+1j c1k1 e1 + O jk2 j= 1 jc1j2j1j2k + O jk1 jjk2 j :$+,&4-)+$:1&, k 1 + O c11 2 12 (k+1) x(k) )1 jc1j2j1j2k + O jk1 jjk2 j(k) (x1 = (x(k) x(k) ) =jc1j2j1j2k + O jk1 jjk2 j = 1 1 + O c11 2 12 k 0 k 1= 1 + O @1 2 Ajjjj11-$&%(31&, k 1 2(k)c1 k1 e1 + O j2 jc1 1 x(k )e1 = kx(k) k = k k 1=2 = k 1=222kjc1j j1j + O j1 jj2 j1 + O c11 21 0 k 1= ei'e1 + O @ 2 A 1 k%,+ ei' = cc11 11 . &'+*- ,&2->-1-.kjc1 k1c1 1 k e1 + Ojjjjjj..jjjjj x5.
& '1+ %854 1. 9'( '+-$(>-0(( 1- '+-$:1&8 F ()+'-0(&11<8 7'&0+// (1)/6&,()/@, ,-A+ +/$( ./$&4(+ (x(0) e1 ) 6= 0 1+ 4<7&$1+1&. +$& 4 )&*, 3)& (>->7'(/.)/)4(@ &2'.%$+1(8 ,$@ 1+2&)&'&%& k0 =.,+) 4<7&$1+1& (x(k0 ) e1 ) 6= 0, ,-A++/$( (x(0) e1) = 0. 9&/$+ ;)&%& &7(/-11<8 4<K+ 7'&0+// '-=&)-+), 2-2 +/$( =<+%& 1-3-$( / x(0) = x(k0 ) .4 2.
/$( j1j > 1, )& kx(k) k = c1k1 e1 + O jk2 j! 1 7'( k ! 1 T/$( j1j < 1, )& kx(k) k = c1k1 e1 + O jk2 j ! 0 7'( k ! 1 . 9&;)&*., 3)&=< 1+7'&(>&K$& 7+'+7&$1+1(@ ($( 7&)+'( )&31&/)(, 1-,& 3+'+> 2-2&+-)& 2&$(3+/)4&m ()+'-0(8 1&'*('&4-): 4+2)&' x(k) )-2, 3)&=< kx(k) k = 1. 9'-2)(3+/2( ;)&&/.C+/)4$@+)/@ )-2: +/$( k ,+$()/@ 1- m 1-0+$&, )& 7&/$+ 4<3(/$+1(@ x(k+1) ((k+1)(1k) 7&$-%-+* x(k+1) = kxx(k+1) k .
-$++ 7'&0+// 7'&,&$A-+)/@, 2-2 +/$( =< *<+%& 1-3-$( / x(0) = x(k+1) .x 4.2. 6 "& % 5%6 $ 8*"*%$@ 4<7&$1+1(@ K-%- -$%&'()*- (1) ().+. 4<3(/$+1(@ 4+2)&'- x(k+1) ( 4+$(3(1< (1k) )'+=.+)/@ 4<3(/$()::1) 4+2)&' x(k+1) = Ax(k) T ,$@ ;)&%& )'+=.+)/@ n2 + O(n) -,,()(41<6 ( /)&$:2&A+ *.$:)(7$(2-)(41<6 &7'+-0(8 (,$@ 4<3(/$+1(@ 7'&(>4+,+1(@ *-)'(0< A 14+2)&' x(k) T2) /2-$@'1&+ 7'&(>4+,+1(+ a = (x(k) x(k) )T ,$@ ;)&%& )'+=.+)/@ n + O(1) -,,()(41<6 ( /)&$:2& A+ *.$:)(7$(2-)(41<6 &7'+-0(8T3) /2-$@'1&+ 7'&(>4+,+1(+ b = (x(k+1) x(k) )T ,$@ ;)&%& )'+=.+)/@ n + O(1)-,,()(41<6 ( /)&$:2& A+ *.$:)(7$(2-)(41<6 &7'+-0(8T4) &)1&K+1(+ 1 = a=b T ,$@ ;)&%& )'+=.+)/@ 1 *.$:)(7$(2-)(41-@ &7'+-0(@..**('.@ ;)( &0+12(, 1-6&,(*, 3)& 1- &,(1 K-% -$%&'()*- )'+=.+)/@ n2 +O(n) -,,()(41<6 ( /)&$:2& A+ *.$:)(7$(2-)(41<6 &7'+-0(8.x5.
# +)&, 4'-C+1(8 2&=( 7&>4&$@+) 1-6&,(): 4/+ /&=/)4+11<+ >1-3+1(@ /(**+)'(31&8 4+C+/)4+11&8 *-)'(0< A 2 Mn .x 5.1. 5 "*%5%$". $@ 4/@2&8 *-)'(0< B = (bij ) 7&$&A(*Y(B ) =nXiji=jjbij j26.. x5. & '1+ %86{ /.**- 24-,'-)&4 41+,(-%&$:1<6 ;$+*+1)&4 *-)'(0< B .9./): A = (aij ) 2 Mn(R), A = A = At . /@2-@ /(**+)'(31-@ 4+C+/)4+11-@*-)'(0- ,(-%&1-$(>('.+*- 4 1+2&)&'&* +42$(,&4&* =->(/+, ).+. /.C+/)4.+) &')&%&1-$:1-@ *-)'(0- Ob 2 O(n) ( ,(-%&1-$:1-@ *-)'(0- X = diag(1 : : : n) )-2(+, 3)& A = Ob XOb . )/L,- X = Ob AOb = OAO , %,+ &=&>1-3+1& O = Ob 2 O(n).3+4(,1&, 3)& ,$@ ,(-%&1-$:1&8 *-)'(0< Y(X) = 0 ( ,$@ 4/@2&8 *-)'(0< BY(B ) 0.
$+,&4-)+$:1&-) Y(OAO ) 0 ,$@ 4/@2&8 O 2 O(n)T=) Y(OAO ) = Y(X) = 0, +/$( O = Ob .$+,&4-)+$:1&, *-)'(0- O = Ob @4$@+)/@ '+K+1(+* >-,-3( *(1(*(>-0((5.120(&1-$- Y(OAO ) 1- %'.77+ &')&%&1-$:1<6 *-)'(0:Y(OAO ) ! Omin:O(n)2/$( *< 1-8,+* 2-2&+-)& '+K+1(+ O1 ;)&8 >-,-3(, ).+. Y(O1 AO1 ) = 0, )& *-)'(0- Y(O1 AO1 ) ,(-%&1-$:1- ( &')&%&1-$:1& 7&,&=1- *-)'(0+ A .
$+,&4-)+$:1&,1- ++ ,(-%&1-$( /)&@) (/2&*<+ /&=/)4+11<+ >1-3+1(@ *-)'(0< A ..,+* /)'&(): 7&/$+,&4-)+$:1&/): /(**+)'(31<6 *-)'(0A = A0 A1 : : : Ak : : :(1))-2.L, 3)& ,$@ 4/@2&%& k = 1 2 : : : :1) $+,.LC-@ *-)'(0- &')&%&1-$:1& 7&,&=1- 7'+,<,.C+8 (( 7&)&*.
&')&%&1-$:1& 7&,&=1- (/6&,1&8)Ak = Ok Ak 1Ok Ok 2 O(n):(2)2) .**- 24-,'-)&4 41+,(-%&1-$:1<6 ;$+*+1)&4 /$+,.LC+8 *-)'(0< /)'&%&*+1:K+ /.**< 24-,'-)&4 41+,(-%&1-$:1<6 ;$+*+1)&4 7'+,<,.C+8 *-)'(0<:Y(Ak ) < Y(Ak 1)(3)).+. 7&/$+,&4-)+$:1&/): fY(Ak )gk=1 /)'&%& *&1&)&11& .=<4-+), 3)& 4 /($.Y(Ak ) 0 %-'-1)('.+) /.C+/)4&4-1(+ 7'+,+$- klim Y(Ak ).3) F)&) 7'+,+$ '-4+1 1.$L:lim Y(Ak ) = 0:(4)k;;1!1!1-)'(0< Ok 2 O(n) 4 (2) 7&,=('-L)/@ 1- K-%+ k )-2, 3)&=< .,&4$+)4&'():./$&4(@* (3), (4).% 1.
( " > 0 { . * (1) | (4) k = k0 , Y(Ak0 ) < " . ( & A i , 1 i n , , j ; a(iik0 ) j <pn ; 1p" ,(a(ijk0 ) ) { &Ak0 . , pp Ak0 ' n ; 1 " A ... x5. & '1+ %87:"#&. /($. (4) ,$@ 4/@2&%& " > 0 /.C+/)4.+) k = k0 , )-2&+, 3)&0 Y(Ak0 ) < " . /($. (2) 4/@2&+ /&=/)4+11&+ >1-3+1(+ *-)'(0< A @4$@+)/@/&=/)4+11<* >1-3+1(+* *-)'(0< Ak0 .9& )+&'+*+ !+'K%&'(1- ,$@ 4/@2&%& { /&=/)4+11&%& >1-3+1(@ *-)'(0< Ak0/.C+/)4.+) i , 1 i n , )-2&+, 3)&j ; a(iik )j Ri(Ak ) =000nXj =1j =ivvnun uXuja(ijk0)j uut 12uuutX ja(ijk0)j2j =1j =i6j =1j =i66 pn ; 1pqpY(Ak0 ) < n ; 1 "+&'+*- ,&2->-1-.
*+)&,+ 4'-C+1(8 2&=( 2-3+/)4+ &')&%&1-$:1<6 *-)'(0 Ok 4 (2) (/7&$:>.L)/@ *-)'(0< ;$+*+1)-'1&%& 4'-C+1(@ (/*. /)'. 43): Ok = Tijk = Tij ('k )."%&$ ' 7&,=('-+)/@ )-2, 3)&=< .,&4$+)4&'(): (3), - (1,+2/< i ( j { )-2, 3)&=<.,&4$+)4&'(): (4).x 5.2. '>% !*" &%?@<3(/$(* ,$@ 7'&(>4&$:1&8 /(**+)'(31&8 *-)'(0< A ( 7'&(>4&$:1&8 *-)'(0< ;$+*+1)-'1&%& 4'-C+1(@ Tij *-)'(0. B = Tij ATijt ( 4<'-A+1(+nXY(B ) ; Y(A) =lm=1l=m(b2lm ; a2lm):69'( .*1&A+1(( A 1- Tij /$+4- (>*+1@L)/@ )&$:2& /)'&2( i ( j *-)'(0< A , 7'(.*1&A+1(( Tij A 1- Tijt /7'-4- (>*+1@L)/@ )&$:2& /)&$=0< i ( j *-)'(0< Tij A .9&;)&*. 4/+ ;$+*+1)<, 1+ 1-6&,@C(+/@ 4 /)'&2-6 i , j ( /)&$=0-6 i , j , .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
