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CI1l(dng3 : Nghi~m Chinh Boa Cua Pht(dng Trlnh Tich Phan Lo1;liMQL CHUaNG 3: NGHIEM CHiNH HOA CUA PHl!(JNG TRINH TicH )\ )\ PHAN LOJj.I MQT Trong chuang Lase gidi thi~u mQts6 phuong phap chinh hoa cho phuong trlnh Au =g vdi A la loan tit tuye'n tinh,li = A*Ra.(AA* ) , dilt AA* = ARa(A.)A* = A*Ra(A-) => Ra(A.)A*g E RangeA* Da ne'u gQi {UI1,VI1, Pn} la singular system cl'ta A(xemO.7) Taco: R a (A)A.*g = (~ n.- )(A *g, v n )vn = R Av } ( n~l a ~n )~' n n ~ R n =1 a 00 l- ( = ~ R ~l-2 _ n- a n g,~V )( v ~n n] n ( ) = ~ ~l~1Ra ~~ ~, vn} n n =1 (~~ 2)~, vn} n R (A)A*g=lim ~ ~l ~1-2R ( )(g,V}=n n=l~ ~n ~,un } n a n=l n n ann = n= ~ 1~n~~2Ra lim a~O ~l-2 Trang 27 (Do (2) , (3) va dinh ly hQit~lbi ch~n.) =A+g (do 2.7.4) Trong tru'ong hQp g~D(A +) ke't qua san chi r~ng (Ra (A)A"g) akhong hQi t~l(ngay d hQi t9 ye'u) 3-1-2 Dinh Iv : Ne'u g~D(A +)khi m6i day an + O;(Ran(A)A"g)nla khong hQi t~ Clui'l1J? 111;1111 - G9i Pia phep chie'u tntc giao tilY len Range(A) = Ker(A"l=Ker (A).L w Ra n (A)A'Pg = Ra (A)A'g +z EX n w =>A'Ra => ARa n n (A)Pg +z=>AA'Ra(A~g +AZ n (A)rg +Az (Do A la compact xemO.9) l~i (2) va (3) ta co: Til SHYfa Pg V~y: w (Ran ARa n (A)Pg +Pg =Az => g E D(A +) mall n +oo thuffn gia thie't g ~ D(A +) (i'\jA' gIn kh6ng h~i II! y(!n 3-1-3 He Qua: Ntu g ~ D(A~) thl lim liRa(A) A*gll = 00 ' A a +0 ~ ,!! Ha1 d ~nh Iy tren ch1 rang de t11Udu'QCslj h91 t~ cua xa := Ra(A)A"g thl di€u ki~n dn va du la g E D(A+)hay Pg E RangeA (P la phep chie'u trljc giao cua Y leu RangeA) Bay giOta khao sat t6c dQhQit~lcua Xav€ ' ' ? A' ? A+g Ta hoan loan co the thay the' di€u ki~n (3) : tRaCt) bi ch~n dell bai di€u ki~n I tVll-tRL(t)l~w(a,v) Voi tE I (4) [o,[IAW]a day w (a, v) la ham so' (chi t6c dQ clia slj hQi t~l) thoa marT w(a ,v) +0 a +0 cho m6i v > 3-1-4 Eli'd~ : Range (AV) ~ Ker (A).L Clu(l1g l11inl1 Avx = I vnn(x,w~ n= I n (xemO.7) d day Anla gia tri rieng khac ct1aA va wIllavectd rieng tu'dngling , -1 - "-1 " Do : Wn= An A Wn= A An Awn E Range A -v " L => A x E RangeA = (KerA) (xemO.6.3) 3-1-5.Dinh iv : Ntu AAv x =Pg VOlmQtv > va x EX nao thl : II A+g -x CIUfl1J? D~t u 111;1111 a II ~ w (a, v) II xii - =A+g Au = Pg =AA v x => A (ll - AVx) = => II - A v X E KerA Trang 28 U E (KerA) i va A v x E (KerA) i ~ u - AVx E (KerA) i ma : KerA n (KerA) i {O} - vx=O~u=A = -v x ~u-A Xa= Ra (A) A*g = Ra(A)A* Pg ma: Pg= AAvx= Au ~ Ra (A) A* AAVx= Ra(A)A*Au ~Ra(A)A v+l x = Xa do: Ilu- x(x" = II Avx - Ra(A)Av+IXII = II A v ( -Rex(A)A)x II II u -xu II ~ w (a,v) II xii (do (4» Bay giCl d~t ea : =A~g - Xa 3-1-6 Dinh Ii ! Ne'u PgE Range (A V)tuc Pg = A Vxvdi mQt v :2 : lieall ~ fw(a, v-1Xa, v)llxll Clui:n!?minI!: f)~t u = A+g~Au = Pg = A.vx= AA*CA.v-I)X ~A(u - A*A v-IX)= ~u- A*A.v-lx E KerA ll1 nhu dinh 19 tren ta cling duc;Jc : u = A +g = A* A 19-1co : Xa= Ra (A)A*Pg = Rcx(A)A*A x = A*Ra (A)A x lam ttlong v-IX ~v ~ ~v ~Ca = A+g - Xa = U - Xa = A* A.v-IX - A*RaCA)A Vx = A*(I - Ra(A)A.)A v-IX ~ II ea 112= (ea , A*(I-Ra(A)A.)A =(Aea , (I - Ra (A)A)A v-IX) ~ ~ IIcall S II II V-IX) A call II (I - Ra(A.)A)A v-Ixll A ea II II xII w (a , v-I) (*) 19-ico : A ea = AA*( - Ra(A.)A.)A.v-IX= A*A.v(I - RaCA)A.)x Do : II A ea 112 = (A ea , A ea) = (A*A ea, ea) = (A ea , eeL) v = (A*A (I - Ra(A)A)x, ea) v = (A ( - Ra(A)A)x , Aea) ~ ~ ~ ~ ~ ~ II Ae(xI12~IIAV(I-Ra(A.)A)II.IIAeall.llxll sw(a,v).IIAcall.llxll ~IIAeallsw(a,v)llxll NhUV?y(*)thanh:11 cuI12sw(a,v)llxll.llxll ~ s jw(a,v)w(a, v-I) Ileal! (do(4» w(a,v-l) Jlxll Nhu V?y bhng dinh 19 3-1-1 va dinh 19 3-1-2 ta da chI duc;Jcr~ng Ra(A)A *g + A+g va chI g ED(A+) Va m,;lI1h hon nua ne'uPg E Range(A.* V) \:div :2 1th1 Ileal!= IIA+g - Xall ~ jw(a, v -1)w(a, v) Ilxll ~ - v * Trang A x =Pg va Xa= Ra (A)A g Bay giClta chuySn sang xet cho traCInghc;Jpg'6la du ki0 - * '6 E>~t: Xa = Ra (A)A g Ntu X: hQit~ltheo mQtnghlan~lOdo v€ nghi IIA+ g - x ~(o) II -+ + Khi -+ => xa(o) -+ A g Bay giO ta xet de'n khai nit$m chinh hoa ye'u cac di€u kit$n khong m ta c6: v€ A+g -+ Ne'u di€u kit$n khong tho a thl ke't lu~n nhu the' nao v€ s\f hQi tl,lcua va g E D(A+) J =0 la di€u kit$n dll clla s\f hQi tl,lm0 va g'"thoa th6a di€u ki~n ligon - gll ~ n kh6nghqit~ yea Clucnfjminh : GQi {un,VIl,~lll} la singular system cila A va d~HAn=~l~2 (Anla gia tri rieng cL\aA*A, Anla day giam v€ kh6ng VIA kh6ng co d~ng huu Iwn xem 0.7 ) [0,+00) ~[O,+oo); a(O) =0 Ben 38n ~ cho a(On)=All Do di€u ki~n lien t~lCcl\a a: s: 1::\ t 8n u~1 g = g-Onlln ~ Chli y r~ng :Xn(8n) =I~n(81l) (A)A*g ~ A +g n~ 00 (do 3-1-1) Trang 32 L~i co: xa[ 0n) - x a(o n) = R a(o n /A)A * (g - g n ) (A)A* v = Ra(o n ) n n -0 - - - nlln a(o n ~l-IR II aeon) -xon aeon) -02 - 11 -2R nlln n (\ n n a(o) n \/ n f' n => x ) (AfY ~\- -IR R / (/ n f -02/ n n a(on ) ( nf a(o) n - = o~a(on)Ra(o n ) (a(on)f ~ Ta co: Khi n -t 00 xa(o n ) = xa(o n ) -xa(o n ) +xa(o n ) ;:::Ilxa(o n ) Ma: x + va a(o n ) ~ A g =>fIxal(on ) ~ 00 x n a(o n ) 11 - xa(o n ) 00 =>xa(o n ) KhonghOit~lye'u Nhu' v~y dieu ki~n g EOD ( A +) va 11gB - gll ~ - x a(o n ) - "x a(6 n ) +00 s; ta cHikh':l0 sat sl! hOi t\l cua x~(o) ve xa ph~l thuOc VaG Ra(t) Bay giG ta xem m (au '" '" + A Au - A g,v) = EO\Iv EOX '" '" =>au + A Au= A g Sv nha't : Do axa + A"'Axa= A"'Ag(aI + A"'A)xa=A"'gDen ta chi dn chung minh (aI + A"'A)la 1-1 '" '" Ta co : (aI + A A)x = =>(ax +A Ax,x) = => a IIxl12+ ( A'"Ax,x) = => allxJJ2+ IIAxl12 = => IIxii =0 => x = O Ngoai aI + A'"Ala tuye'n tinh V?y aI + A'"A la - Sv phi,!thuQc lien t~!C: Ta chi dn chung minh ne'u ax + A'"Ax = z -> thl x -t Ta co : ax + A"'Ax= z =>(ax + A'"Ax,x)= (z, x) * 2 =>allxll +(A Ax,x)=(z,x)=>allxll +IIAxll-=(z,x)::;llzll.llxll Trang 34 =>aIlxf ~ Ilzll.llxll=>allxll~ Ilzll Cho z -+ => a Ilxll -+ => x -+ O Xu 1a c\fc lieu phie'u ham Fu Ta co: -g112 +allxI12-CIIAXa -g112 +a11Xa112 Fa(X)-Fa(Xa)=IIAXa = flAx - gl12 -IIAX a - gl12 + aCllxf -llx a 112) =IIAX-AXa -I-Axa -g112-IIAxa _gll2 +acIIX-Xa =IIACX-Xa)112 -I-2CAxa -g,ACX-Xa)-I-aIIX-XaIl2 +Xa112-IIXaI12) -I-2a(Xa,X-xa) = IIACX- Xa)112 -I-2CAxa - g,A(x - xa) -I-allx - xal12 -I-2a(Xa'X - xa) * * => Fu (x) - Fu(xu) ;::2(A Axu - A g, x - xu) + 2a( Xu, x - xu) * * => Fu(x) - hxCxu);::2(A Axu -A g + axu, x - xu) ,* * ma : A Axu - A g + axu = => Fu(x) - Fu(xu) ;::O NgliQc l?i ta cling tha'y Fu co day nha't mQt c\fc lieu xac dinh bdi 1"(0) =0 vdi mQi w E X do: [(i) = Fu(xu + tw) = IIA(Xa+ tw - gl12 -I-allXa + tIff , * * [(0) = 2(A Axu - A g + a Xu , w) = V w Do Xu tho a : A*Axu - A*g +a Xu = * => A Axa.+ a Xu= A g Dieu phli hQp vdi Xu= (A -I-a1)-1 A*g Vdi ham chi t6c dQ w(a,v) = av nhli da noi d tn~n Cling \rdi cac ke't qua 3-1-5 va 3-1-6, ta dliQc M qua san : 3-2-2 H~ (jua : Ne'uA+gE Range(A.V)vdi O