HUKUM SYLVESTER INERSIA

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1 Vol 6 No Desember 3 ISSN : HUKUM SYLVESTER INERSIA R Heru Tjhj Jurus Mtemt FMIPA UNDIP Abstr Mtrs represets sutu betu udrt dpt dsj sebg mtrs dgol Eleme pd dgol utm mtrs represets tersebut dpt dpdg sebg ugs ler yg td tuggl Kre td tuggl m dperlu teorem tu huum yg megtur rterss represets yg dpt dsj deg td tuggl Huum lh yg del sebg huum Sylester IersHuum Sylester tetg Iers meyt bl U rug produ dlm rel d xy orm bler smetr d U m terdptlh sutu bss ={ } dr U sedem hgg dlh mtrs dgol deg j = j deg = j << =- j <<r d = j r<< lebh ljut d r tertetu deg tuggl oleh Kt uc: Rug produ dlm orm bler smetr mtrs dgol PENDAHULUAN etu udrt dlm sutu produ dlm dpt drepresets oleh mtrs represets Mtrs represets tersebut dpt dsj sebg mtrs dgol Eleme-eleme o dgol utm dlh eleme ol sedg eleme-eleme dgol utm pd mtrs dgol tu dpt dpdg sebg ugs ler Permslhy dlh ugs ler yg terdpt dlm dgol utm dpt dsj secr td tuggl Kre peyjy td tuggl m dperlu tur tu huum yg megtur blm sutu represets dpt dsj deg td tuggl Huum Sylester Iers mejwb permslh tersebut deg megtur rter represets yg dpt dsj secr td tuggl PEMAHASAN Agr dlm membc tuls seljuty td djump slh pegert berut dtuls dulu beberp des yg perlu dethu Des Fuhrm996 Msl U rug Vetor ts lpg blg omple K Produ dlm U dlh ugs <>: UxUK yg memeuh <xx> > d <xx>= j 44

2 Huum Sylester Iers R Heru Tjhj d hy j x= <x+yz>=<xz>+<yz> <xy>= y x utu setp xyu utu setp K d rug etor U ts K deg produ dlm dsebut rug produ dlm Dber trorms ler T dlm rug produ dlm U ts lpg K ddes eld-lued ucto pd UxUK deg xy=<txy> J memeuh <x +x y>=<x y>+<x y> m dt ler terhdp x J memeuh <xy +y >=<xy >+<xy > m dt ler terhdp y J memeuh <xy +y >= <xy >+ <xy > m dt tler terhdp y J ler terhdp x d tler terhdp y m dsebut ugs sesquler tu orm J ler terhdp x d y m dsebut ugs bler 3 etu udrt dlm rug produ dlm U dlh sutu ugs berbetu ˆ x= xx deg xy memeuh xy=<txy>=<ytx> 4 J ={ } bss rug produ dlm U yg berdmes hgg T trorms ler d U Ddes j =<T j > m mtrs j dsebut mtrs represets dr orm terhdp bss mtrs dber ots Ctt Ctt utu Teorem Spetrl Setdj999 J semu l rterst r m T= E + E + + r E r mtrs A yg smlr deg mtrs r Seljuty j betu udrt pd rug produ dlm U m terdptlh sutu bss ={ } sedem hgg mtrs represets terhdp bss tersebut dlh mtrs dgol hl dsj dlm teorem 3 berut 45

3 Vol 6 No Desember 3 ISSN : Teorem J betu udrt pd rug produ dlm U m terdpt bss ={ } sedem hgg dgol ut : Msl xy betu udrt pd U Teorem dbut deg dus Msl lg dmu= J = m dlh mtrs bertpe x yg merup mtrs dgol Msl teorem terbut utu rug berdmes < - d msl lg betu udrt berdmes d ={ } bss d U A terdpt du sus Ksus pertm dmsl td semu eleme-eleme pd mtrs dlh ol rty terdpt des sedem hgg Tp ehlg umumy but ddes bss bru ={ } deg re merup bss re bebs ler = rty sebg ombs ler dr = - rty sebg ombs ler dr d 3 3 = 3 - = - rty 3 sebg ombs ler dr 3 d rty sebg ombs ler dr d Perht bhw ombs ler dr { } dpt dsj sebg ombs ler { } yg bebs ler Abty { } bebs ler 46

4 Huum Sylester Iers R Heru Tjhj Seljuty eruty perht = - = - = - = - = Secr log 3 = 4 = = = = 3 = = = Abty = o [ M ] o deg M= L d o ={ } erdsr hpotes dus [ M] o o merup mtrs dgol m seluruh terbut mtrs dgol Ksus edu dmsl semu eleme-eleme pd mtrs dlh ol rty utu semu des = Msl = J j =j m dlh orm ol ds dlh mtrs ol d mtrs ol pst mtrs dgol Kt msl terdpt psg j sedem hgg j Tp ehlg umumy but t msl = d j= Utu peyederh ots dtuls = Tuls = + / d = - / m =/ = = - / Seljuty dplh etor-etor l pd bss-bss bru mejd betu = - - utu =345 dperoleh = = sehgg ddpt psg persm sstem persm sebg berut = = = =

5 Vol 6 No Desember 3 ISSN : Sstem mempuy solus = / d = / Serg ddes ={ } deg = + / = - / d = =34 bss U sebb utu setp dpt dyt secr tuggl sebg ombs ler dr =34 Perht bhw = o o M deg o ={ 3 } d M=L 3 erdsr hpotes dus [ M] o o merup mtrs dgol m seluruh terbut mtrs dgol Abt Msl betu udrt yg terdes pd rug produ dlm U ={ }dlh sebrg bss U ={g g } dlh bss yg medgol Msl =g g =Ig g j [x] = [x] = d I = j m xx= J j j

6 Huum Sylester Iers R Heru Tjhj 49 ut : xx=<txx>=< [x] [x] > =< [x] [x] >=< j [x] [x] >= j = t j = t = t = Perht bhw [x] = I [x] = j = Ddpt m J j j Abty xx = = J j j Seljuty j teorem dpeuh m hsl j utu setp dtuls dlm teorem Teorem berut merup Huum Sylester ers

7 Vol 6 No Desember 3 ISSN : Teorem Huum Sylester Iers J U rug produ dlm d xy ugs bler smetr d U m terdptlh sutu bss ={ } ytu bss U sedem hgg dlh mtrs dgol deg j = j deg = r r lebh ljut d r tertetu deg tuggl oleh ut: Kre smetr m terdpt opertor xy=<txy> T d U sedem hgg J dber sebrg bss ortoorml m dlh mtrs smetr Msl ={e e e + e r e r+ e }terdr dr etor-etor rterst yg bersesu deg l rterst + r r+ m = r Perht bhw r = r d r sutu mtrs tuggl Smp d s sudh terbut r tuggl tggl dbut terdpt deg tuggl yg tertetu oleh Ambl = d betu ={ - e r - e r }={ - e r - e r } m 5

8 Huum Sylester Iers R Heru Tjhj = O r O O Meurut teorem Spetrl terjm d deg tuggl Des lg r d s=-r yg tertetu oleh huum Sylester tetg ers berturut-turut dsebut r d sgture dr betu udrt Sgture betu udrt dots deg d sgture mtrs Hermt A dots sebg A Mtrs bujur sgr A ogrue deg j terdpt mtrs bujursgr osgulr R sedem hgg =RAR Abt Du betu udrt pd rug produ dlm U ogrue j d hy j mempuy r d sgture yg sm Du mtrs smetr bertpe x ogrue j d hy j mempuy r d sgture yg sm ut : Deg meggt Teorem bhw betu udrt dpt dwl oleh mtrs dgol x m utu membut bt cuup dbut du mtrs x ogrue j d hy j r d sgture-y sm Dr ljbr ler dethu bhw du mtrs ogrue j d hy j ry sm Kre r sm j d hy j s sm m terbut du mtrs x ogrue j d hy j r d sgture-y sm 5

9 Vol 6 No Desember 3 ISSN : Dlm teorem 3 berut dtuls sutu st yg berlu utu sebrg mtrs smetr A bertpe mxm d X mtrs bertpe mx yg mempuy r brs peuh utu sutu mtrs smetr x yg ddes =XAX m r d sgture A d tept sm Teorem 3 Msl A mtrs smetr mxm X mtrs bertpe mx yg mempuy r brs peuh mtr smetr x yg ddes =XAX m r d sgture A d tept sm ut: Perht bhw dr =XAX ddpt Ker XKer lsy: Ambl sebrg YKer X m XY= Y=XAXY=XAXY=XA= Kre Y= m YKer Lg perht dr =XAX ddpt Im Im X lsy: Ambl sebrg YIm m d Z R sedem hgg Z=Y XAXZ=Y XAXZ=Y Jd Y Im X Seljuty mtrs X dpt dsj sebg X=A X deg A bertpe mx-m d X bertpe mxm sehgg d bss C sedem hgg X= X ertbel XAX= X Perht dulu X A [A] X = X bty r = r A X = deg A bertpe mxm AX X AX sehgg =X AX 5

10 Huum Sylester Iers R Heru Tjhj Dr = = X AX ddpt r = r = r A Abty dperoleh = =A rty teorem terbut Seljuty utu C R d A= m A mtrs Hermt yg ogrue deg =dg jug utu A A Hermt yg memeuh ra +A =ra +ra m A +A =A +A Hl dsj dlm teorem 4 berut Teorem 4 J C R d A= m A mtrs Hermt d A ogrue deg =dg J A d A Hermt d ra + A =ra +ra m A +A = A + A ut : A= hermt sebb = = = Jd A=A Perht bhw mtrs dpt dprts ts brs-brsy tu olomolomy row993 hl t dp dlm lgh pembut seljuty A= = =

11 Vol 6 No Desember 3 ISSN : = + + = = = = =VV Kre A= VV m dsmpul A ogrue deg Msl l-l rterst yg td ol dr A yg berhubug deg etor rterst d msl + + +l l-l rterst yg td ol dr A yg berhubug deg etor rterst + + +l m meurut teorem spetrl

12 Huum Sylester Iers R Heru Tjhj 55 A d A l Perht bhw ra = d ra =l A = : C C A = l : C C Kre dethu r A +A = r A +ra m dm Im A +A = dm Im A +dm ImA berrt dm Im A dm ImA = bty Im A dm ImA ={} Meurut teorem Spetrl { + + +l } bebs ler m A +A = + l = A +A = A +A Ddpt A +A mtrs Hermt Dr A +A = l m meurut butr A +A ogrue deg mtrs dgol + + +l A +A l rty A +A =A +A

13 Vol 6 No Desember 3 ISSN : KESIMPULAN erut dtuls esmpul sebg hsl peelt J U rug produ dlm rel d xy ugs bler smetr dlm U m terdpt sutu bss ={ j } ytu bss U sedem hgg dlh mtrs dgol d j = j deg = r Lebh ljut d r tertetu r deg tuggl oleh DAFTAR PUSTAKA rowwc Mtrces Oer Commutte Rgs Mrcel Deer New Yor 993 Fuhrm A Polyoml Approch to Ler Algebr Sprger-Verlg New Yor Setdj Dtt Aljbr Ler Ljut F MIPA Uersts Gdjh Md Yogyrt