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leqcia 4

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leqcia 4

kalkulusi sabumenismetyvelo mevnierebebisaTvis 07.05.2010

    leqcia 4

    3. wrfivi algebruli gantolebebi da veqtoruli algebra 3.1. veqtorebi, matricebi, determinantebi. pirveli da meore rigis kontaqtebi epidemiologiaSi

    gansazRvra 3.1.1. n namdvili ricxvisgan Sedgenil simravles, romelic garkveuli rigiT Cawerilia

    striqonSi an svetSi, Sesabamisad, ewodeba veqtor-striqoni an veqtor-sveti:

    y??1????xxx?y???x:,,, . ??y:??1n?? 2??xy??n

    x-s da y-s ewodebaT x da y veqtorebis i-uri ii

    komponentebi.

    ?nulebisgan Semdgar veqtors nulovani veqtori ewodeba: x?(x,x)12 2 x0?????, . (,,)00?????0??

    O x yoveli organzomilebiani (orkomponenta) veqtori x 11???x:?x,x(x,x) SeiZleba ganisazRvros sibrtyis werti-1212

    lis saSualebiT (ix. nax. 3.1.1); es veqtori gamoisaxeba

     nax. 3.1.1 isriT, romlis saTave emTxveva koordinatTa sistemis

    xx,saTaves, xolo bolo _ wertils. ??12

     2x

    ???y?(y,y)x?y?(x?y,x?y)121122

    ?x?(x,x)12 O x 1

    nax. 3.1.2

    ??yyy??,,?,,gansazRvra 3.1.2. da veqtorebis jami ganisazRvreba Semdegi xxx?????1n1n

    tolobiT (ix. nax. 3.1.2): ?????z?x?y:?x?y,?,x?y. 11nn

    1

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    ??,,gansazRvra 3.1.3. c skalarisa da veqtoris namravli ganisazRvreba Semdegi to-xxx???1n

    lobiT ?cxcxcx. :,,????1n??yyy??,,?,,gansazRvra 3.1.4. ori da veqtoris skalaruli (Siga) namravli xxx?????1n1nganimarteba Semdegi tolobiT:

    n????x,y:?xy???xy?xy. ?11nnkk?1k?piTagoras Teoremis Tanaxmad (ix. nax. 3.1.1) veqtoris sigrZis kvadrati x

    ???222, xxxxx???,??12

    xx,xolo sigrZe (SegveZlo pirdapir migveRo koordinatTa sistemis saTavesa da wertils Soris ??12

    manZilis formulis gamoyenebiT)

    ???22. :,xxxxx?????12??,,analogiuri formulaa samarTliani n-ganzomilebiani veqtorisTvis: xxx???1n

    ???22. ????:,xxxxx???1n

    mn?gansazRvra 3.1.5. matrici ewodeba garkveuli rigiT dalagebul raodenobis ricxvebisgan Sedgenil marTkuTxa cxrils, romelic m striqonisa da n svetisgan Sedgeba:

    ??aaaa??111211jn????aaaa212222jn??

    ??????????A:?. aaaa????iiijin12

    ????????????aaaa????mmmjmn12

    aam ricxvebs matricis elementebi ewodebaT. -Ti aRniSnulia i-uri striqonisa da j-uri svetis ij

    gadakveTaze mdebare elementi.

    moxerxebulobisTvis A matrici mokled ase Caiwereba

    . Aaa????ijij

    cxadia, veqtor-striqoni (veqtor-sveti) da () matrici erTmaneTs emTxveva. 1?nn?1

    gansazRvra 3.1.6. matrics, romelic miiReba mocemuli A matricis striqonebis SecvliT imave

    TAnomris svetebiT an piriqiT, ewodeba transponirebuli da aRiniSneba -Ti:

    aaaa????112111im??aaaa??122222im??

    ????????T. A:???aaaa????jiijmj12

    ??????????aaaa????nninmn12

    ab?or erTnairi ganzomilebis Aa? da Bb? matrics toli ewodeba, Tu . ijijijij

    2

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    n striqonisa da n svetisgan Semdgar kvadratul matrics n-uri rigis matrici ewodeba. mis , aii

    , elementebs diagonaluri elementebi ewodeba. Tu matricis diagonaluri elementebi 1-is, in?12,,,?

    xolo danarCeni elementebi 0-is tolia, maSin mas igivuri I matrici ewodeba. gansazRvra 3.1.7. erTnairganzomilebiani ori marTkuTxa da matricis jami da Aa?Bb?ijij

    c skalarisa da matricis namravli ganisazRvreba Aa?ij

     da A?B:?a?bcA:?caijijijtolobebiT.

    mn?gansazRvra 3.1.8. ganzomilebis matricisa da ganzomilebis Aa?np?Bb?ijij

    matricis namravli ganisazRvreba

     CABc??ijtolobiT, sadac

    n

    c:?ab:?ab?ab???ab. ?ijikkj1i1ji22jinnjk1?

    csxva sityvebiT, warmoadgens A matricis i-uri striqonisa da B matricis j-uri svetis skalarul ij

    namravls.

    cxadia, mxolod iseTi matricebis gamravleba SeiZleba, roca samravlis svetebisa da mamravlis

    striqonebis raodenoba erTmaneTs emTxveva.

    aRsaniSnavia, rom kvadratuli matricebisTvis samarTliania namravlis asociaciuroba

    , magram komutaciuroba, sazogadod, irRveva . Tu ()()ABCABC?ABBA?

    1211????, , A?B?????1111????maSin

    121133111223????????????, . AB??BA??????????????111122111123????????????magram

    3323???? ?????2223????da, amdenad,

    . ABBA?

    gansazRvra 3.1.9.

    aa??1112 A???aa??2122

    matricis determinanti Semdegnairad aRiniSneba

    aa1112 AA??detaa2122da ewodeba

    detA:?aa?aa 11222112ricxvs.

    analogiurad, mesame rigis

    3

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    aaa111213

     aaa212223

    aaa313233

    determinanti ewodeba

    aaa?aaa?aaa?aaa?aaa?aaa 112233213213311223312213113223211233

    ricxvs, romelic gamoiTvleba samkuTxedis (ix. nax. 3.1.3) an sarusis (Sarrus) wesiT (ix. nax. 3.1.4).

    nax. 3.1.3.

     +

    *)n-uri rigis A matricis n-uri rigis determinanti ewodeba wevris jams, sadac: yoveli wevri A n!matricis n elementis namravlia, amasTan, matricis yoveli striqonidan da yoveli svetidan determinan-

    tis wevrSi Tanamamravlad aRebulia erTi da mxolod erTi elementi; determinantis wevri aiReba `+~

    an `-~ niSniT imisda mixedviT, Tanamamravlebis svetebis (striqonebis) nomrebisgan Sedgenili

    mimdevroba naturalur ricxvTa 1,2,?,n mimdevrobisgan wevrTa luwi raodenobis ricxvjer gadasmiT (transpoziciiT) miiReba Tu kentiT, roca Tanamamravlebi dalagebulia striqonebis

    (svetebis) nomrebis zrdis mixedviT 1-dan n-mde. **)SemoviRoT levi-Civitas

    i...i1...2i...i1n1n?? ?????1...n...iii...i1n1n

    i,?,isimboloebi. isini +1-is (Sesabamisad -1-is) tolia, Tu indeqsTa mimdevroba miiReba 1n

    1,2,?,nnaturalur ricxvTa mimdevrobis wevrTa luwi (Sesabamisad kenti) raodenobis ricxvjer

    gadasmiT; sxva SemTxvevaSi (e. i. roca indeqsebidan ori indeqsi maincaa erTmaneTis toli) isini

    nulis tolia.

    advili misaxvedria, rom Tu es simboloebi nulisgan gansxvavebulia, maSin misi nebismieri ori

    indeqsis gadasmisas isini niSans icvlian. magaliTad,

    i...i...i...ii...i...i...i1lkn1kln????

    iiam tolobaSi gadasmulia da indeqsebi, xolo sxva indeqsebi Tavis adgilzea datovebuli. kl

    determinantis zemoT moyvanili ganmartebidan gamomdinare, cxadia,

    aa?111nnnnniiii......ii?44nn1n. (3.1.1) ??????a...a?????a...a?a?a????iinini1ini11111nnn?1?1iiii?1?11n1naa?1nnn

    AgansazRvra 3.1.10. determinantis romelime elementis minori ewodeba determinants, romelic miiReba im striqonisa da svetis amoSliT, romlebsac mocemuli elementi ekuTvnis.

    AaM determinantis elementis minors aRniSnaven simboloTi. ikik

     *)nn ( faqtoriali) niSnavs 1-dan -amde naturaluri ricxvebis namravls, e. i. n!

     n!:?1?2???n.amasTan vTvliT, rom 0!?1.**) levi-Civita tulio (1873 _ 1941) _ italieli maTematikosi da meqanikosi.

    4

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    + aaa111213 + aaa???212223aaaaa+ aaa1112131112 313233 aaaaa aaa2122232122111213 aaaaaaaa3132333132212223 ???

    nax. 3.1.4

    AagansazRvra 3.1.11. determinantis elementis algebruli damateba ewodeba mis minors, ik

    ik?()?1Agamravlebuls -ze. mas aRniSnaven simboloTi, e. i. ik?ikA:?(?1)M. ikiknebismieri rigis determinants gaaCnia Semdegi Tvisebebi:

    determinantis sidide:

    1) ar Seicvleba, Tu mis striqonebs Sesabamisi svetebiT SevcvliT, e. i. transponirebuli determinantebi tolia;

    2) icvlis niSans, Tu adgils SevucvliT or striqons an or svets;

    3) izrdeba k-jer, Tu romelime striqonis (svetis) elementebs gavamravlebT k-ze, e. i. romelime

    striqonis (svetis) elementebis saerTo mamravli SeiZleba gavitanoT determinantis niSnis gareT;

    4) nulis tolia, Tu ori striqonis (svetis) Sesabamisi elementebi proporciulia, kerZod tolia;

    5) Tu misi romelime striqoni (sveti) Sedgeba ori Sesakrebis jamisgan, ori determinantis jamis tolia,

    romelTagan erT-erTSi jamis nacvlad pirveli Sesakrebia, xolo meoreSi _ meore;

    6) ar Seicvleba, Tu romelime striqonis (svetis) elementebs mivumatebT sxva striqonis (svetis)

    Sesabamis elementebs, gamravlebuls erTsa da imave ricxvze;

    7) romelime striqonis (svetis) elementebis maT algebrul damatebebze namravlTa jamis tolia; kerZod,

    nulis tolia, Tu romelime striqonis (svetis) yvela elementi nulia. yvela es Tviseba mtkicdeba (3.1.1) formulis gamoyenebiT.

    me-7 Tvisebas ewodeba determinantis daSla romelime striqonis (svetis) elementebis mixedviT.

    es ukanaskneli me-3 rigis determinantisTvis striqonis mimarT ase Caiwereba

    3

    A:?aAi?1,2,3, , ?ijij1j?

    an gaSlili saxiT:

    AaAaAaA???;?111112121313? AaAaAaA???;?212122222323

    ?AaAaAaA???.313132323333?

    determinantis daSla xelsayrelia im striqonis (svetis) elementebis mixedviT, romelic met nulebs

    Seicavs.

    SeniSvna 3.1.12. romelime striqonis (svetis) elementebis sxva striqonis (svetis) Sesabamisi

    elementebis algebrul damatebebze namravlTa jami nulis tolia.

    AgansazRvra 3.1.13. kvadratul A matrics ewodeba gadagvarebuli, Tu misi determinanti nulis

    tolia, winaaRmdeg SemTxvevaSi A matrics ewodeba aragadagvarebuli.

    5

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    ?1gansazRvra 3.1.14. aragadagvarebuli A matricis Sebrunebuli matrici ewodeba matrics, Aromelic akmayofilebs

    ??11 AAAAI??

    tolobebs, sadac I erTeulovani matricia.

    determinantebis me-7 Tvisebisa da SeniSvna 3.1.12-is gamoyenebiT advilad davaskvniT, rom

    ??AAA112131??AAA??

    ??AAA?1122232A?, ??AAA??AAA??132333??AAA??

    Asadac aris a elementis algebruli damateba. ikik

    gansazRvra 3.1.15.

    AAA??112131?? AAA??122232??AAA??132333matrics matricis mikavSirebuli matrici ewodeba. A???a:?(a,a,a)b:?(b,b,b)Tu da mocemuli veqtorebia, maSinñ veqtoris veqtoruli a123123?

    bnamravli -ze gamoiTvleba ?? ????????a?b:?ab?ba;ab?ba;ab?ba232331311212formuliT.

    es formula me-2 rigis determinantebis meSveobiT Semdegi saxiT Caiwereba:

    ???aaaaaa?233112ab:;;??. ??bbbbbb233112?????????????a?ae?ae?aeexxeeñ da veqtorebis, sadac, , Sesabamisad , , b?be?be?be11223331122 112233x RerZebis mimarTulebis mqone erTeulovani veqtorebi _ ortebia, veqtoruli namravlis povna 3

    moxerxebulia Semdegi

    ???eee123? a?aaa123

    bbb123formulis gamoyenebiT.

    aa?gansazRvra 3.1.16. n-uri rigis a matrics ewodeba simetriuli (antisimetriuli), Tu ijjiij

    a??a() jn?12,,,?. ijji

    matricebis gamoyenebis TvalsazrisiT ganvixiloT epidemiologiaSi pirveli da meore rigis kontaqtebis amocana. vTqvaT, 3 adamiani daavadda infeqciuri daavadebiT da mimdinareobs 6

    adamianisgan Semdgari jgufis gamokiTxva avadmyofTa (pirvel) jgufTan kontaqtebis dadgenis

    mizniT. garda amisa, xdeba 7 adamianisgan Semdgari mesame jgufis gamokiTxva maTi meore

    jgufTan kontaqtebis TvalsazrisiT. SemoviRoT ganzomilebis Aa? matrici Semdegnairad: 36?ij

    6

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    Tu meore jgufis j-ur adamians hqonda kontaqti pirveli jgufis i-ur adamianTan, maSin CavTvaloT,

    a?1a?0rom , winaaRmdeg SemTxvevaSi (e. i. maT kontaqti ar hqoniaT) CavTvaloT, rom . ijij

    b?1analogiurad SemoviRoT ganzomilebis matrici da CavTvaloT, rom , Tu Bb?67?ijij

    mesame jgufis j-ur adamians hqonda kontaqti meore jgufis i-ur adamianTan, xolo winaaRmdeg

    b?0SemTxvevaSi CavTvaloT, rom . es ori matrici aRwers pirveli rigis kontaqtebs jgufebs Soris. ij

    magaliTad, vTqvaT, A da B matricebs aqvT Semdegi saxe

    0010010????0011000??001010????1000011??, . A?100100B?????0011000????001101????0101000??

    1000010??

    a?1gansaxilvel SemTxvevaSi niSnavs, rom meore jgufis meoTxe adamians hqonda kontaqti 24

    b?0pirveli jgufis meore avadmyofTan. niSnavs, rom mesame jgufis mesame adamians ar hqonia 33

    kontaqti meore jgufis mesame adamianTan.

    Tu Cven gvainteresebs arapirdapiri kontaqtebi (e. w. meore rigis kontaqtebi), e. i. mesame jgufis

    adamianebis kontaqtebi pirveli jgufis sam avadmyofTan, maSin pasuxs miviRebT A da B matricebis

    gamravlebiT:

    . CcAB??ij

    elementi

    6cab? ?ijikkjk?1

    gvaZlevs mesame jgufis j-uri adamianis pirveli jgufis i-ur avadmyofTan arapirdapiri kontaqtebis ricxvs. Cvens SemTxvevaSi

    1101011????. CAB??0021010????2011021??c elementi gviCvenebs, rom mesame jgufis mesame adamians hqonda meore rigis 2 kontaqti 23

    pirveli jgufis meore avadmyofTan. SevniSnoT, rom mesame jgufis meeqvse adamians pirveli jgufis

    pirvel, meore da mesame avadmyofTan hqonda Sesabamisad 1, 1 da 2 meore rigis kontaqti, e. i.

    pirveli jgufis avadmyofebTan, saerTo jamSi, hqonda meore rigis kontaqti. mesame 1124???jgufis mexuTe adamians ki meore rigis kontaqtebi avadmyofebTan saerTod ar hqonia.

    racionTa matricebi. vTqvaT, sakvebi Seicavs cxovelis sicocxlisTvis aucilebel nivTierebas,

    ?magaliTad, A vitamins da -Ti aRvniSnavT am vitaminis raodenobas 1 kg i-ur sakvebSi i

    . maSin veqtor-striqoni (,,,)in?12????a:??,?,?,? 12n

    xiZleva vitaminis ganawilebas sakvebebis mixedviT. vTqvaT, cxoveli Wams pirveli sakvebis kg-s, 1

    xxme-2 sakvebis kg-s da a.S. n-uri sakvebis kg-s. rogor gamovTvaloT A vitaminis raodenoba, 2n

    romelsac cxoveli dReSi iRebs?

    7

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    ????x:?x,x,?,xcxadia, amisTvis sakmarisia, veqtori skalarulad gavamravloT veqtorze a12n??(a,x)??x??x??????x. 1122nnskalaruli namravli SegviZlia ganvixiloT, rogorc matricTa namravlis kerZo SemTxveva, rogorc

    n?1 ganzomilebiani a matricis gamravleba ganzomilebiani X matricze. matricTa gamravlebis 1?n

    wesis gamoyenebiT gveqneba

    x??1??x??2

    ?????. ?????,?,...,???x??x?????x?12n1122nn???

    ???????xn??

    marjvena nawilSi miviReT erTelementiani matrici. misi elementis sidide gviCvenebs A vitaminis raodenobas, rasac cxoveli erT dReSi miiRebs.

    axla ganvixiloT ufro rTuli SemTxveva. vTqvaT, gvainteresebs sami A, B, C vitaminis cxovelis mier miRebis sakiTxi. vTqvaT, 3?n ganzomilebiani

    ??????12n?? M?????12n??????????12n

    matrici gviCvenebs sakvebSi vitaminebis raodenobas. yoveli striqoni Seesabameba calkeul vitamins,

    ?xolo yoveli sveti _ calkeul sakvebs. amrigad, pirveli sakvebis 1 kg Seicavs raodenobis A vita-1

    ??mins, raodenobis B vitamins, raodenobis C vitamins da a. S. vTqvaT, matrici-sveti x11

    1x????2x??

    ??? ???x???

    ???????x??n

    dRe-RameSi sakvebis moxmarebis maCvenebelia. rom gavigoT cxovelis mier dRe-RameSi

    miRebuli vitaminebis raodenoba, unda vipovoT M da X matricebis namravli.

    x??1???x??x?...??x????????1122nn12n????x??2???????MX????x?x...?x????. ??12n1122nn????????????x??x?...??x12n1122nn??????xn??

    miviReT matrici-sveti, romlis I elementi aris A vitaminis, II elementi _ B vitaminis da III elementi _ C vitaminis saerTo raodenoba, rasac cxoveli dRe-RameSi iRebs.

    ngansazRvra 3.1.17. amovarCioT -uri rigis determinantis striqoni da sveti, ; ?kk1?k?n?1

    ganvixiloT determinanti, romlis elementebic amorCeuli striqonebisa da svetebis gadakveTaze

    8

kalkulusi sabumenismetyvelo mevnierebebisaTvis leqcia 4

    *)mdgomi elementebia. miRebul determinants determinantis -uri rigis minori ewodeba. Tu ?MkamovSliT amorCeul striqons da svets, maSin darCenil rigis determinants -uri rigis kkn?kk~minoris M damatebiTi minori ewodeba. ~s:?(?1)AMsgansazRvra 3.1.18. minoris algebruli damateba ewodeba sidides, sadac aris M

     minorSi nawilobriv Semavali determinantis striqonebisa da svetebis nomrebis jami. M?

    nlaplasis Teorema. Tu -uri rigis determinantSi amovarCevT striqons ( svets), maSin ?kk

    amorCeul striqonSi ( svetSi) moTavsebuli yvela -uri rigis minoris saTanado algebrul kkkdamatebebze namravlTa jami determinantis tolia. ?

     *)1?k?min{m?1,n?1}min{???} -uri, ( aRniSnavs minimalurs figurul frCxilebSi miTiTebul sidideebs k

    m?nSoris), rigis minori analogiurad ganimarteba matricisaTvis da is ewodeba determinants, romelic Sedgeba matricidan amorCeuli striqonisa da svetis gadakveTaze mdgomi elementebisagan. kk

    9

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