
khote bijganitacha niyamit ghatak

ganitat, khote bijganit kinwa khote gatacha niyamit ghatak haah eka ghatak aahe jyache kendrikarak pariman shakya titke lahan aahe.
udaharanarth, eka guntaguntichya ardh-sadhya khote bijganitemadhye, sootra _ 2 madheel tyachya kendrikaranache pariman sootra _ 2 chya rankchya barobariche asalyas ghatak sootra _ 1 niyamit asate, jeey kahee carton up-bijganit sootra _ 4 chya parimanasarkhe asate (lakshat ghya qee aadhichya kagadpatranmadhye, jatil ardh-sadhya khote bijganitachya ghatkala jarr too ardh-sope asel tarr niyamit mhatale gele hote aanee tyachya sanlagn pratinidhitvatil curnell haah carton up-bijganit aahe).
jarr tyachya kendrikaranache pariman sootra _ 6 chya shrenichya barobariche asel tarr ghatak sootra _ 5 eka khote gut niyamit aahe.
moolbhut prakaran.
sootra _ 7 chya vishisht prakaranat, bijganitiyadrushtya bund kshetra sootra _ 9 varr sootra _ 8 matrixche khote bijganit (udaharanarth jatil sankhya), niyamit ghatak sootra _ 10 haah eka ghatak aahe jyachya jorden samanya swarupat pratyek eigen mulyasathi ekach jorden block asato (dusya shabdant, pratyek eigen mulyachi bhaumitik gunakar 1 aahe).
matricks sootra _ 10 varr moolyamapan kelelya sootra _ 11 peksha qamee degreechya bahupadicha sanch haah niyamit ghatakacha kendrikarak asato aanee mhanoon kendrikaranache pariman sootra _ 11 asate (jeey sootra _ 14 chya shrenichya barobariche asate, parantu teey bijganitiy torrus asane aavashyak nahee).
jarr matricks sootra _ 10 karnabaddh asel, tarr teey niyamit asel jarr aanee sootra _ 11 bhinn eigenvalues asateel tarach. heey pahanyasathi, lakshat ghya qee sootra _ 10 konatyahi matricks sootra _ 18 saha pravas karel jeey tyachya pratyek eigenspacela sthir karate. jarr sootra _ 11 bhinn aaygan moolye asateel, tarr heey tevhach ghadate jevha sootra _ 18 heey sootra _ 10 chya samaan aadharavar tiryak asel; kharey tarr sootra _ 18 heey sootra _ 10 chya pahilya sootra _ 11 shaktinche resheey sanyojan aahe aanee kendrikarak heey jatil pariman sootra _ 11 (vastavik pariman sootra _ 26) chey bijganitiy torrus aahe; kaaran heey kendrikaranache sarvat lahan sambhavya pariman aahe, matricks sootra _ 10 niyamit aahe. tathapi, jarr samaan eigenvalues asateel, tarr kendriyak heey sootra _ 10 chya igenspaceschya samanya resheey gatanche utpadan aahe aanee tyache pariman katekorpane mothe aahe, jenekarun sootra _ 10 niyamit nahee. 
jodalelya sankshipt khote gut sootra _ 30 sathee, niyamit ghatak eka khula ghan upasanch tayar karatat, joou kamal torrus sootra _ 32 madheel ghatakanchya sootra _ 30-sanyog varganni banalela asato, jeey sootra _ 30 madhye niyamit asatat. sootra _ 32 chey niyamit ghatak swatah sootra _ 32 madheel sanchache purak mhanoon spashtapane dile jaatat, jeey sootra _ 30 chya mool pranalishi sambandhit codimention-vann subtoricha sanch aahe. tyachpramane, sootra _ 30 chya khote bijganit sootra _ 2 madhye, niyamit ghatak eka khula ghan upasanch tayar karatat jyache spashtapane varnan sootra _ 32 chya khote bijganitachya niyamit ghatakanche sanlagn sootra _ 30-orbits mhanoon kele jaau shakate, jeey mool pranalishi sambandhit hyperplane bahereel ghatak aahet.
vyakhya.
sootra _ 2 heey anant kshetravaril maryadit-aayami khote bijganit asoo dya. pratyek sutrasathi _ 42, chala
sootra _ 45 chya sanlagn endomorphism sootra _ 44 chee vaishishtyapurn bahupadi asso. mug, vyakhyenusar, sootra _ 2 chee shreni heey sarvat qamee purnank sootra _ 47 aahe jase qee kahee sootra _ 49 sathee sootra _ 48 aanee sootra _ 50 dware darshavile jaate. udaharanarth, pratyek "x" sathee sootra _ 51 asalyane, sootra _ 45 heey nilpotant (i.eae.jarr aanee fakt sootra _ 54 asel tarr, pratyek sootra _ 53 heey engalchya prameyanusar nilpotant aahe.
sootra _ 55 asey samajoo. vyakhyenusar, sootra _ 2 chaa niyamit ghatak haah sanch sootra _ 57 chaa eka ghatak aahe. sootra _ 58 heey sootra _ 2 vareel bahupadi karya asalyane, zariski topologychya sandarbhat, sanch sootra _ 57 haah sootra _ 2 chaa khula upasanch aahe.
sootra _ 62 varr, sootra _ 57 haah eka jodalela sanch aahe (nehamichya topologychya sandarbhat), parantu sootra _ 64 varr, too fakt jodalelya khulya sanchancha maryadit sangh aahe.
eka carton up-bijganit aanee eka niyamit ghatak.
anant kshetravar, eka niyamit ghatak carton up-bijganit, eka swayam-samanyikrut nilpotent up-bijganit tayar karnyasathi vaparla jaau shakato. vaishishtyapurn shoonyachya kshetravar, haah drushtikon serve carton subulgebra tayar karato.
ghatak sootra dile _ 42, chala
eigenvalue shunyasathi sootra _ 53 chey samanyikrut eigenspace vha. heey sootra _ 45 chey up-bijganit aahe. lakshat ghya qee sootra _ 69 heey sootra _ 53; i chya aaygan moolya mhanoon shoonyachya (bijganitiy) gunakarasarkhech aahe.eae., sarvat qamee purnank "m" asey aahe qee sootra _ 71 madhye sanketan aahe. asha prakare, sootra _ 72 aanee samanata jarr aanee fakt sootra _ 73 haah niyamit ghatak asel tarach lagoo hote.
asey vidhan aahe qee jarr sootra _ 73 haah eka niyamit ghatak asel tarr sootra _ 75 haah eka carton up-bijganit aahe. asha prakare, sootra _ 76 heey kiman kahee carton up-bijganitache pariman aahe; kharey tarr, sootra _ 76 heey carton up-bijganitache kiman pariman aahe. adhik jordarpane, vaishishtyapurn shoonyachya kshetravar (udaa.jee.sootra _ 64 kinwa sootra _ 62),
eka guntaguntichya ardh-sadhya bijganitachya carton up-bijganitetil eka niyamit ghatak.
jarr aanee jarr too hyperplane sootra 90 chya sanyogat nasel, tarr, carton up-bijganit sutrasathi, karanah sootra 91 sathee, sootra 86 chaa ghatak niyamit asato, jarr too jarr hyperplane sootra 90 chya sanyogat nasel tarr, jarr too mool pranali sootra 88 saha, eka jatil ardh-sope bijganit sootra 45 asel, tarr sootra 86 chaa ghatak niyamit asato.
heey vaishishtyikaran kadhikadhi niyamit ghatakachi vyakhya mhanoon ghetale jaate (visheshtah jevha carton subalgebramadhil keval niyamit ghatak swarasyapoorn asatat).