Analisis Diagonalisasi Matriks untuk Menentukan Individu ke-n Berdasarkan Peluang Genotip Induk

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1 98 BoWll Jurl Ilm Ilmu Bolo M 5 Vol. No., p 98-3 ISSN: -6 Alss Dolss Mtrks utuk Mtuk Ivu k- Brsrk Plu Gotp Iuk M. Yk Slm K, Mmk Ujt Rom, Prorm Stu Mtmtk FMIPA Urm Jl. Mjpt 6 Mtrm 835. Tlp Eml : mmk_ur@yoo.om Gtk l sl stu r lmu olo y mmpljr rts ytu puru st r stu rs k rs rkuty. Sr prkm lmu ptu pr plt mptk sutu vu y uul r rp rs sutu vu. Plt rtuju utuk mtu plu r sutu vu jk lkuk prkw rs k-. Prmsl trsut pt slsk lss olss mtrks rsrk plu r otp uk sutu spss. Hl-l y rkt utuk mlkuk olss l l vktor r mtrks plu otp uk. Hsl y prol rup prsm x = A x, mk pt ktu plu vu p rs k- ktk lmt muju tk uk BLL rup.5 BBLL,.5 BLL,.5 LL. Prtu lm plt muk plks MATHEMATICA 6. Kt-kt ku : Dolss mtrks, lmt, otp. PENDAHULUAN Gtk l sl stu r lmu olo y mmpljr rts ytu puru st r stu rs k rs rkuty. St y wrsk r sutu uk rup st otp otp. Gotp l tuk tu susu ts sutu krktr y ku vu lm sl tuuy, sk otp l sutu krktrstk y pt ukur tu st y yt p sutu orsm. Krktrstk tu s tmpk sprt wr ukur (Wllm.99). Dlm olo muul sutu prmsl ytu mtu sutu vu y uul r rp rs jk lkuk prkw sr kotyu trp vu trsut, mu jk lkuk sr kovsol st mruk kr wktu y ukup lm y y t jk spssy trolo lk ml. Prmsl trsut pt slsk lss olss mtrks rsrk plu r otp uk sutu spss. Mtrks y uk rsl r tl plu otp uky Plt rtuju utuk mmut sstm prsm lr rsrk tl otp prkw. Brsrk tl y k trtuk plt pt mtu st vu p rs k- r lss olss mtrks. Prtu p plt muk Mtmt vrs 6 utuk mmprmu. Apu rp s y uk l s rkut. Ds : Sutu prsm lr lm pu (vrl) l prsm tuk : x + x + 3 x x =,, 3, l l rl x, x, x 3, x l pu. D mk sutu sstm lr r m prsm lm pu l sutu sstm rtuk : x + x + + x = x + x + + x = m x + m x + + m x = m m m (utuk m, =,,..., ) l l rl s prsm ts s sstm prsm lr (m ). Btuk sstm prsm lr trsut pt u mj sutu mtrks, murut Ato () tuk mtrks r sutu systm prsm lr vrl m prsm sr umum l s rkut :

2 [ m m m3 x x m ] [ x ] = [ m ] Akty rsrk prsm trsut kt pt mulsk rturut-turut s mtrks A m, x, m, mk sstm prsm lr trsut sr umum pt tuls Ax =. Solus r prsm lr x + x + 3 x x = l sutu urut r l s, s,..., s smk rup s prsm trsut k trpu jk kt mtk x = s, x = s,..., x = s. Kumpul smu solus r prsm tu sut mpu solus tu solus umum. Stp sstm prsm lr pt tk mmlk solus, mmlk tpt stu solus, tu mmlk tktr yky solus. Prsm Ax =, mk sstm prsm lr omo jk l =, jk sut sstm prsm lr tk omo. Ds : Mslk A l sutu mtrks. Sklr λ sut s sutu l tu l krktrstk r A jk trpt sutu vktor tkk ol x, s Ax = λx. vktor x sut vktor tu vktor krktrstk r λ. METODE PENELITIAN Dlm plt mto y lkuk l stu ltrtur. Lk prtm ytu mmut tl plu rsrk otp uk y pl, lm l uk y uk l BLL smu kmuky. Lk rkuty ytu mlkuk pross olss mpulsy. Brkut l ljut k s p pms. HASIL PENELITIAN Pross Dolss Pross olss mtrks l s rkut : Mslk A l mtrks.. Ttuk l r mtrks A ytu λ, λ,..., λ =,,.. Ttuk vktor r l y trkt mtrks A syrt stp vktor-vktor p, p,..., p l s lr. 3. Dtuk sutu mtrks P p, p,..., p s vktor-vktor kolomy.. Mtrks P AP l mtrks ol λ, λ,..., λ s tr-tr oly. Brsrk mtrks olss D = P AP k lkuk mpuls ljr utuk mr rumus mtrks D A. Mslk D = P AP Utuk mr D mk : D = P APP AP Prtk P P = PP = I mk D = P AAP D = P A P Utuk mr D 3 mk : D 3 = P A PP AP Prtk P P = PP = I mk D 3 = P A AP D = P A 3 P S utuk D mj : D = P A(PP )A(PP )AP P AP D = P A P Brsrk sl D = P A P, k ttuk mtrks A ytu s rkut : D = P A P Ku rus k klk P s mj : D P = P A PP Prtk P P = PP = I mk D P = P A Sljuty ku rus kr klk P mj : PD P = PP A Prtk P P = PP = I mk PD P = A () Tl Plu otp r prkw BLL smu kmuky Got p Gotp Iuk Kturu BBLL- BLL BBLl- BLL BBll- BLL BLL- BLL BLL- BLl BLL- Bll BLLLL BLLLl BBLL BBLl BBll BLL BLL -ll

3 BLl Bll LL Ll Bll Dlm mtuk sl kturu p rs = otp LL p rs k- k- k mslk tuk l r otpotp kturu s rkut : = otp Ll p rs k- = otp BBLL p rs k- = otp BBLl p rs k- = otp BBll p rs k- = otp BLL p rs k- = otp BLl p rs k- = otp ll p rs k = Utuk =,,,... Brsrk tl () trsut k rumusk sutu sstm prsm lr ytu = otp Bll p rs k- = = = = = = = = = Brsrk sstm prsm lr trsut pt u lm tuk mtrks x = Ax sr rturut-turut s rkut :

4 [ ] [ ] 8 [ ] D =,,... Brsrk prsm x = Ax mk k lkuk mpuls s rkut : = A(Ax ) = A (Ax 3 ) = A (Ax ) = A x mk x pt Dsk s : x = A x Utuk mr l A k lkuk pross olss mtrks rsrk pross olss. Lk prtm ytu mr l r mtrks A. Brsrk Ds () ytu l mtrks A pt r t(a Iλ) = S ptk Prtk x = Ax x = Ax 3 mk x = Ax t(a Iλ) = λ λ6 6 7λ7 + 9λ8 λ9 = Mk prol kr-kr r polom rrjt sml s rkut : λ =, λ =, λ 3 =, λ =, λ 5 =, λ 6 =, λ 7 =, λ 8 =, λ 9 = S mtrks A pt Dsk s rkut: A = CD C Sljuty kt pt mtuk tuk r x = A x D vktor x = (,,,,,,,, ) D sk sutu tuk : = utuk =,,,... r.

5 3 3 x

6 3 D mk utuk mtu plu otp vu p rs k- mk pt lt r lmt muju tk utuk mtrks x. lm x prol sl s rkut : Tl Hsl Ivu P Grs k- No Brs Lmt Hsl Ktr { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } BBLL BBLl BBll BLL BLl Bll LL Ll Bll Brsrk sl otp y prol p rs k-, pt rk sutu stu ksus trp j sutu tumu utuk mlt st otp y muul p rs k- kt st otp y trj. Tl 3 Gotp Fotp No Gotp Fotp BBLL Smoot Yllow BBLl Smoot Yllow 3 BBll Smoot Gr BLL Smoot Yllow 5 BLl Smoot Yllow 6 Bll Smoot Gr 7 LL Wrkl Yllow 8 Ll Wrkl Yllow 9 Bll Wrkl Yllow Brsrk tl trsut pt ktu sl vu p rs k-, ytu vu rotp BBLL plu ssr.5 y mmlk st j Smoot Yllow, BLL plu ssr.5 mmlk st Smoot Yllow LL plu ssr.5 y mmlk st j Wrkl Yllow, sk tumu ly mmlk l plu otp ol p rs k-. Hl rrt smu vu y muul mmlk st j y ukup k. KESIMPULAN. Sutu prmsl tk ytu pwrs st rsrk plu otpy pt sjk lm tuk sstm prsm lr x =. Apu y mmpru sstm prsm lr trsut l st r uk-uky.. Mtrks plu otp ukur 9x9 mmlk vktor-vktor y s lr s pt lkuk pross olss. 3. Hsl vu y trtuk trp uk BLL l.5 BBLL,.5 BLL,.5 LL. st-st otp y l yk vrt r st-st tumu y trtuk l yk. Brsrk plt y tl lkuk rpk pt jk s u utuk pr plt tk slum mlkuk pro prkw sutu tumu u st. Drpk ju plt y mmprlus plt pt mlkuk moks st-st otp y l yk vrt r st-st tumu y trtuk l yk. DAFTAR PUSTAKA Ato, H. & Rors, C.,, Aljr Lr Elmtr Vrs Aplks, Es 8, Jl. Erl. Jkrt. Brtl, Rort G. & Srt, Dol R. 97. Itrouto to Rl Alyss. Jo Wsly & sos. Nw York. Wllm D. 99.Tor Sol-Sol Gtk.Erl. Jkrt. Wjyt, K Prp Dolss Mtrks Dlm Gtk Trp. Ckrwl Pk. XVI :93-. Yul, S.. Prp Dolss Mtrks Mtrks Lsl Dlm Mmproyksk Juml Populs Prmpu. UNNES Jourl o mtmts., ():5-5