HANDOUT MATRIKS & RUANG VEKTOR

KULIAH HANDOUT MATRIKS & RUANG VEKTOR. DEFINISI MATRIKS MATRIKS dh kmp ig-ig yg diss secr khss dm etk ris d koom sehigg eretk empt persegi pjg. St mtriks A yg terdiri dri m ris d koom dpt ditisk segi Am. t A(m ). Beerp Jeis Mtriks Berdsrk Ss Eemey. Mtriks kdrt t mtriks jr sgkr. Mtriks o 3. Mtriks digo 4. Mtriks st 5. Mtriks skr 6. Mtriks tridigo 7. Mtriks qsi-digo 8. Mtriks segitig wh d segitig ts 9. Mtriks simetris. Mtriks skew. Mtriks skew simetris

Bc-cetk Vektor Bris Progrm kompter dm QUICK BASIC tk memc-mecetk vektor ris: DIM C(5) ================================== NRD = INPUT DEVICE CODE NUMBER NWR = OUTPUT DEVICE CODE NUMBER NAMA PROGRAM : VEX.BAS ================================== CLS NRD = OPEN DATA7.DAT FOR INPUT AS #NRD FOR I = TO 5 INPUT #NRD, C(I) NEXT I CLOSE (NRD) NWR = 3 OPEN DATA8.DAT FOR OUTPUT AS #NWR FOR I = TO 5 PRINT #NWR, C(I); ; NEXT I CLOSE (NWR) END

3 KULIAH. OPERASI DENGAN MATRIKS. PENJUMLAHAN d h mtriks hy didefiisik pi ked mtriks yg dijmhk it sejeis. D h mtriks diset sejeis i kr kedy sm. Bi A m. + B m. = C m., dm h ii eeme-eeme dri mtriks C m. dh: c ij = ij + ij tk i =,,, m j =,,,. Sprogrm Srotie Pejmh Mtriks: Digrm ir: SUBROUTINE SUMN (M, N, A, B, C) DIMENSION A(M,N), B(M,N), C(M,N)

4 START I =, M J =, N C(I,J) = A(I,J) + B(I,J) RETURN

5. PENGURANGAN Mtriks mtriks C m. dh: Bi A m. - B m. = C m., dm h ii eeme-eeme dri c ij = ij - ij tk i =,,, m j =,,,. Hkm-hkm yg erk pd pejmh mtriks, erk jg pd pegrg mtriks..3 PERKALIAN Mtriks D h mtriks A d B is dikik pi jmh koom mtriks A sm deg jmh ris mtriks B. Dm etk mm dpt ditisk: c ij = k ik. kj deg i = ide ris =,,, m j = ide koom =,,,.

6 Sprogrm Srotie Perki Mtriks: START I =, M J =, L C(I,J) = K =, N C(I,J) = C(I,J) + A(I,K) * B(K,J)

7 RETURN.4 PERKALIAN MATRIKS DENGAN VEKTOR KOLOM Dm etk mm dpt ditisk: c i j ij j tk i j,,,m,,,. Srotie Perki Mtriks deg Vektor Koom START I =, M C(I) = J =, N C(I) = C(I) + A(I,J) * B(J)

8 RETURN KULIAH 3 3. PERKALIAN VEKTOR BARIS DENGAN MATRIKS Adik dikethi Vektor ris (mtriks ris) X = ( m ) mtriks A m m m Perki tr kedy dpt dikerjk i jmh koom dri mtriks yg pertm sm deg jmh ris dri mtriks yg ked. Dm etk mm dpt ditisk: Y i = j j ji tk i =,,, j =,,, m.

9 Srotie Perki Vektor Bris deg Mtriks START I =, N Y(I) = J =, M Y(I) = Y(I) + X(J) * A(J,I) RETURN

3. PEMBAGIAN DENGAN MATRIKS Istih pemgi deg mtriks tidk egit poper. Utk memgi mtriks A deg B dikk deg cr segi erikt: Utk mecri A C dikerjk C = A.B -, deg B - dh B ivers dri mtriks B. Didefiisik B.B - = I deg I dh mtriks st. 3.3 DEKOMPOSISI MATRIKS St mtriks A dpt didekomposisi mejdi mtriks segitig wh (L) d mtriks ts (U), yit A = LU. Dekomposisi terset ik i digo tm mtriks L t U erhrg st. Misk kit mempyi mtriks A(4 4), mk: 3 4 4 3 43 4 4 34 44 3 4 4 43 44. 3 4 4 34 Yg k dicri dh eeme-eeme dri mtriks L d U. Betk mm dpt dirmsk segi erikt: L i = i tk i =, L ji = ji - i k tk i =, jk. ki j =,

KULIAH 4 3. DETERMINAN 3. Cr Srrs Utk mtriks A 3 3, determiy mert Srrs dicri segi erikt: Det (A) = ( + 3 + 3 ) - ( 3 3 + + ) 3. Cr Mior d Kofktor Det (A) = ij i K ij j =,,, tk >. Kofktor K ij dpt dicri deg mempergk mior M ij K ij = (-) i+j M ij M ij dh mior dri koefisie A ij yg merpk ii determi seteh ris ke i d koom ke j dri mtriks A dihigk.

3.3 Metod CHIO tk Meghitg Determi Adik kit igi mecri ii determi dri st mtriks Mert Chio dekomposisi determi di ts mejdi s-determi erderjt dpt dikk segi erikt: D 3 3 3 3 3 3 3 3 =,,, d setersy.

3 KULIAH 5 3.4 Perhitg Determi deg Opersi Bris Eemeter Perhitg deg memftk kompter tk mtriks kr esr isy tidk memki cr mior d kofktor kre jmh opersiy demiki yk. Metod tk progrm kompter dh:. Uh determi it mejdi determi segitig wh t ts deg meggk opersi ris eemeter.. Nii determi segitig ts (wh) dh hsi perki dri sr-sr digoy. Ctt : Jik dri st determi dikk pertkr ris mk ii determi terset erh tdy. Misk mtriks 3 4 4 3 43 4 4 34 44 deg opersi ris eemeter diredksi mejdi mtriks segitig ts ' 3 ' " 4 ' 4 " 34 ''' 44 Mk: Det (A) = ( ) ( ) ( ) ( 44 )

4 3.5 Perhitg Determi deg Dekomposisi LU Adik A = LU 3 3 3 3 Srrs: Determi dri mtriks L d U di ts dpt dicri deg cr Det L =.. d Det U =.. = Det A = det (LU) = det L. det U =.. Api mtriks A erorde mk Det A =....

5 KULIAH 6 4. PERSAMAAN LINIER SIMULTAN 4. Betk mm st persm iier simt orde N dh: + + + = + + + = + + + = Api sem hrg,,, =, persm terset diset persm iier simt yg homoge. Dm etk mtriks, peis persm iier simt di ts dpt disederhk mejdi: t disigkt mejdi: AX = B. D h khss yg hrs diperhtik dh:. SPL simt mempyi peyeesi tgg i mtriks A dh reger t o sigr yki Det (A).. SPL simt mempyi peyeesi yg tk tet t tk mgki diseesik, i mtriks A dh sigr, yki Det (A) =.

6 4. Peyeesi Persm Liier Simt deg Metod Crmer Api peh dri st SPL orde dh: X j tk j =,,, d persm iier simt terset ditisk dm etk mtriks segi: AX = B mk mert metod Crmer j Det ( A) j Det( A) dm h ii: Det( A) Det j (A) dh determi yg didpt deg meggti koom ke j dri Det (A) deg vektor koom B. Jdi: Det ( A) j.........

7 KULIAH 7 4.3 Peyeesi Persm Liier Simt deg Eimisi Gss Metod eimisi Gss mempergk opersi ris eemeter tk meghpsk sem eeme-eeme mtriks yg erd di seeh kiri digo tm mtriks A( ). Dm peks metod ii, mtriks A( ) ii dijdik A( +) kre vektor koom dietkk di dm koom +. Secr simois. Metod eemisi Gss ii dpt ditergk segi erikt: Misk st persm iier simt, ditisk dm etk persm mtriks segi erikt: Utk mecri hrg-hrg,,,, mtriks egkp diredksi sehigg hsi khiry mejdi ' ' ' Dri mtriks terkhir ii diperoeh:

8 Deg stitsi mdr ertrt-trt diperoeh ii-ii -, -,,,. 4.4 Peyeesi Persm Liier Simt deg Gss-Jord Lgkh-gkh yg dikk dh: Betk mtriks A( ) mejdi A( +) deg meetkk vektor koom pd koom ke + mtriks A( +). Deg opersi ris eemeter, mtriks terset diredksi sehigg dihsik etk terkhir mtriks terset dh: Dri hsi terkhir ii, sdh dpt disimpk hw: = / = / = / Metod Gss d Gss-Jord k erfgsi deg ik i pivot ii dh hrg eeme yg teresr dm ris ke-i.

9 KULIAH 8 4.5 Peyeesi Persm Liier Simt deg Metod Gss- Seide Metod Gss-Seide ii sgt cocok tk peyeesi mtriks erkr esr, t yg yk mempyi eeme erhrg o terserk (Sprse). Cr memki metod Gss-Seide Tisk SPL + + + = + + + = + + + = dm etk segi erikt: = (/ )( - - 3 3 - - ) = (/ ) ( - - 3 - - ) - = (/ - )( - - -, - - -,- - - -, ) = (/ ) ( - - - -,- - ) Kemdi dikk terk dri ii w, misy: X () =, () () =,, =. Sstitsik ii-ii w it ke SPL etk terkhir, didpt X () = /, = = = Sstitsik ii-ii ii ke ris etk terkhir, didpt X () = (/ ) ( - ( / ))

Demiki setersy smpi didpt X () = (/ )( - ( / ) -,- - () ) Berikty proses itersi ke-, ke-3,, smpi ke- yky itersi yg dimit. 4.6 Mtriks Tridigo d Agoritm Thoms Agoritm Thoms sgt cocok tk meyeesik persm iier simt yg dpt dietk mejdi mtriks tridigo. Persm semcm ii yk dijmpi dm perhitg merik persm diferesi prsi deg metod ed erhigg tp eeme erhigg. Misk persm mtriks: AX = B t.... 43... 34 44.. 3 4 3 4 (*) Mtriks yg pig kiri hy mempyi hrg di tridigo, sedgk eeme-eeme di r it erii o. Vektor koom X(,, 3, 4 ) dikethi. Peyeesi (*) dpt dikk deg cr medekomposisi mtriks tridigo A mejdi: A = LU t

.... 43... 34 44.. =.... 43.. 44....... 34.. (**) Api ked mtriks di rs k dikik k didpt etk mm ij = ij = ji tk i =, j =, - ii = ii - ij ji tk i =, j = i-, - ij = ji / ii tk i =, - j = i+, Utk meyeesik persm (**) tereih dh hrs didefiisik vektor koom Y y y y y 3 4 yg memehi persm: LY = B.

K diseesik k didpt etk mm Y = / Y i = ( i - ij y j ) / ii tk y i =, j = i-, - Kre B = LY mk didpt AX = LY LUX = LY t UX = Y.

KULIAH 9 4.7 Peyeesi Persm Liier Simt deg Cr Dekomposisi Tij persm iier simt: AX = B t 4 4 4 4 44 4 4 Dekomposisi mtriks A mejdi perki tr mtriks segitig wh (L) d mtriks segitig ts (U): A = LU yg m L = 3 4. 4. 43 44 d U = 3. 4 4 34 Utk meyeesik persm mtriks AX = B didefiisik st vektor koom Y yg memehi persm LY = B sehigg persm it dpt ditisk LUX = LY t UX = Y Persm LY = B dm etk mm dpt ditisk rms rekrsi segi erikt: y y i i i ii i ii ( i ij j y tk i j ) tk i,

4 4.8 Persm Liier Simt Homoge Betk mm persm iier simt homoge: + + + = + + + = + + + = Peyeesi yg memehi persm di ts dh: = = = = dimk peyeesi trivi. Persm terset mgki mempyi peyeesi yg tidk trivi pi jmh persm eih sedikit dripd jmh peh yg k dicri.

5 KULIAH 5. MATRIKS INVERSI Ivers mtriks A dh A - sehigg memehi A. A - = A -. A = I Utk mecri jmh opersi, dpt dipki cr i yki mecri mtriks iversi deg trsformsi eemeter. Bi A dh st mtriks persegi o sigr kr, mk: A I k dpt ditrsformsik mejdi I A - Misy deg mempergk opersi ris eemeter. 5. Iversi dri Mtriks Segitig Bwh Betk mmy: ii = / ii ij = - ii ( i k j ik kj ) tk i dri smpi tk i = smpi j = smpi. 5. Iversi dri Mtriks Segitig Ats Eeme-eeme ij dpt dirmsk secr eih mm d dpt dikeompokk mejdi 4 grp.

6 Grp pertm: ii = / ii tk i =, N. Grp ked: ij = - ij ij / ii tk i =, N- j = i + Grp ketig: ij ii j k ik kj tk i =, N- j = i+, N Grp keempt: ij ii j k ik kj tk i =, N-3 j = i + 3

7 KULIAH 5.3 Mecri Mtriks Iversi deg Metod Dooitte Ad d mcm pemfktor A mejdi LU yit:. Metode pemfktor Dooitte. Metode pemfktor Crot Pemfktor Dooitte, mesyrtk eeme digo L semy d eeme digo U tko. Misk tk mtriks A() dpt ditis segi 3 3 3 3 Utk meys goritm pemfktor Dooitte perhtik ri erikt ii. 3 3 3 3 3 3 3 3 3 3 Mert kesm mtriks, kit peroeh: I. ; ; 3 3 3 3 3 3 3

8 II. 3 3 III. Agoritm Pemfktor Dooitte Mskk :, ij, i, j =,,,. Ker : L, U Lgkh-gkh: I. Utk j =,,, j j jj Utk i =, 3,, Utk j =,,, i- Sk Utk k =,,, j- Sk sk + ik. kj ij ( ij - sk)/ jj Utk k = i, i+,, Sk Utk m =,,, i - Sk sk + im. mk ik ik - sk II. III. LY = C deg sstitsi mj UX = Y deg sstitsi mdr

9 5.5 Mecri Mtriks Iversi deg Metod Crot Pemfktor Crot, mesyrtk eeme digo L tko d sem eeme digo U erii. Misk tk mtriks A() dpt ditis segi 3 3 3 3 Peys goritm pemfktor Crot dikk seperti peys goritm pemfktor Dooitte. 5.6 Mecri Mtriks Iversi deg Metod Choesky Metod Choesky sgt ermft tk mecri iversi dri mtriks simetris deg eeme-eeme digo tm erhrg positif. Metod ii jg memftk tekik dokomposisi A = LU, k tetpi kre tk mtriks simetris A = A T mk LU = (LU) T t LU = U T L T yg errti: L = U T d U = L T. Jdi dekomposisi A = LU = L L T mk A - = (L L T ) - = (L T ) - L - t A - = (L - ) T L -

3 Dm etk mm metod Choesky dpt ditisk i = i / tk i =, ij [ ij ij j k i i ik jk ] tk i = 3, j =, - ii ii i ik k tk i =,

3 KULIAH 6. MATRIKS TRANSPOSE DAN MATRIKS ADJOINT Adik dikethi mtriks A 3 3 Trspose dri mtriks A dh A T 3 3 Adjoit dri mtriks A ditis Adj (A), dh st mtriks yg eeme-eemey terdiri dri trspose dri sem kofktor eemeeeme mtriks A. Mtriks djoit hy didefiisik tk mtriks kdrt. Adj ( A ) k k k 3 k k k k k k 3 Dm h ii: k = (-) + Det(M ), M dh mior dri koefisie k = (-) + Det(M ), M dh mior dri koefisie d setersy. Perhitg mtriks djoit deg kosep di ts krg efisie kre jmh opersiy ckp esr. Kosep i yg eih ik dh deg mempergk mtriks iversi. Kre tk setip mtriks kdrt erk tr: A. Adj (A) = Det(A).I

Mk dpt ditisk Adj (A) = Det (A).A - Digrm Air Trspose Mtriks START I =, N J =, M ATRAN(J,I) = A(I,J) RETURN

KULIAH 3 Digrm Air Perhitg Mtriks Adjoit START CALL DETMIN (N,A,DET) CALL MATIN (N,M,A,AINVER) I =, N J =, M ADJA(I,J) = DET*AINVER(I,J) RETURN

34 Sprogrm Srotie Adjoit SUB ADJOIN (N, A, ADJA) ======================================== CONTOH SUBPROGRAM SUBROUTINE ADJOINT NAMA PROGRAM : ADJOINT.BAS ======================================== DIM A(N,N), ADJA(N,N) SUBROUTINE LAIN YANG DIPERLUKAN: DETMIN, MATIN CALL DETMIN(N, A, DET) CALL MATIN(N, M, A, AINVER) FOR I = TO N FOR J = TO M ADJA(I, J) = DET * AINVER(I, J) NEXT J NEXT I END SUB

35 KULIAH 4 7. AKAR KARAKTERISTIK 7. Nii Krkteristik (Hrg Eige) d Vektor Krkteristik (Vektor Eige) Nii krkteristik dpt dirtik segi st ii (skr) yg erpsg deg vektor X d memehi persm mtriks: AX = X ( * ) Dm h ii: A dh mtriks kdrt X dh vektor krkteristik Kedy: d X diset kr-kr krkteristik. Persm ( * ) dpt ditisk segi: [ A - I ] [ X ] = [ ] (**) Meys Persm Krkteristik deg Metod Le Verrier- Fddeev Mert teorem Newto, tk st mtriks A jmh dri hrg-hrg eige sm deg jmh dri eeme-eeme digo yg dimk trce t Spr dri mtriks A. Jdi: Trce (A) = Dpt ditis: i i ii. i

36 S S i i i i trce (A) trce (A ) S k i k i k trce (A ) Utk mempermdh perhitg sejty Fddeev megemgk cr segi erikt: A = A ; Trce (A ) = P ; B = A - P I A = AB ; Trce (A ) = P ; B = A - P I A = AB - ; Trce (A ) = P ; B = A - P I Persm krkteristiky: (-) [ - P - - P - - - P ] = 7. Hrg Eige d Vektor Eige dri Mtriks Ordo 3 Hrg Eige Tij persm iier simt segi erikt: + + 3 3 = + + 3 = 3 + + 3 = 3 t AX = B

37 Dm perso mecri hrg eige mk B = I yg m I = sehigg mejdi [ A - I ] [ X ] = [ ] yg merpk persm iier simt homoge. Persm it dpt dietk mejdi 3 3 3 (@) Nii determi [ A - I ] memetk poiomi erderjt 3 dm : 3 + + + 3 = Vektor Eige Tis (@) dm etk ' 3 ' 3 ' 3 Kre peyeesi o-trivi hy dpt dierik dm etk perdig : : 3 mk kit dpt memiih tereih dh hrg w, misy: = sehigg persm it erh mejdi + 3 3 = - + 3 = - + 3 = - 3

38 Didptk: ' 3 3 ' 3 3 3 ' ' Nii-ii,, d 3 dm h ii mejdi vektor eige yg erhg deg hrg eige.