
eka shaktishali

ganitat, ring "r" chaa eka-shakti ghatak "r" assa aahe qee "r" −1 haah eka nirgun ghatak aahe; dusya shabdant, ("r" −1) "n" haah kahee "n" sathee shoonya aahe.
visheshtah, jarr aanee keval jarr tyachi vaishishtyapurn bahupadi 'pii' ('tii') hii 'tii'-1 chee ghaat asel, tarr chauras matricks 'emm' haah eksandh matricks aahe. asha prakare eksandh matrixchi serve eigen moolye 1 aahet. 
ardh-ekshakti yaa shabdacha arth assa aahe qee kahee shakti ekshakti aahe, udaharanarth igenvalues karneey matrixsathi jeey serve ekteche mool aahet. 
bijganitiy gatanchya siddhantanusar, jarr gut ghatak vishisht naisargik gut pratinidhitvat ekatmapane karya karat asel tarr too ekatma asato. eka ekshakitman sanlagn bijganitiy gut haah serve ghatak ekshakitman asalela gut asato.
vyakhya.
matricks vyakhya.
karnakar sutrasah _ 2 saha varchya trikoni matrixchya gut sootra _ 1 chaa vichar qaraa, tyamule teey matrixcha gut aahet.
mug, eka-shakti gatala kahee sutrancha upasamuh mhanoon paribhashit kele jaau shakate. yojana siddhant vaparoon gut sootra _ 1 chee vyakhya gut yojana mhanoon kelee jaau shakate.
aanee jarr hii yaa yojanechi bund gut yojana asel tarr sanlagn gut yojana eksaman aahe.
ring siddhantasaha vyakhya.
jevha tyacha sambandhit yogya anuvad operator, "r" x "," g "chya sanlagn nirdeshank ringvar" a "[" g "] haah ghatak," a "[" g "] chya rekheey endomorphism ringcha ghatak mhanoon sthanik patalivar eksaman asato, tevha sanlagn bijganitiy gatacha ghatak" x "eksandh asato. (sthanik patalivar eka-shakticha arth assa aahe qee "a" ["g"] chya konatyahi maryadit-aayami sthir upkshetravarche tyache nirbandh nehamichya ring-saiddhantik arthane eka-shaktiche aahet.)
jarr tyache serve ghatak ekshaktipurn asateel tarr sanlagn bijganitiy gatala ekshaktiman mhanatat. konatahi eka-shakti bijganitiy gut karn pravishtya 1 asalelya varchya trikoni matrixchya gatachya bund upasamuhasathi samaroop asato aanee yaulat assa konatahi upasamuh eka-shaktishali asato. visheshtah konatahi ekshakti gut haah eka nirbal gut asato, jaree ulat kharey nasale taree (prati-udaharan-jee. ell. 'aine' ('keey') chey karn matricks).
udaharanarth, manak aadhar sutrasah sootra _ 8 varr sootra _ 1 chya manak pratinidhitvamadhye _ 9 nishchit vector sootra _ 10 aahe.
pratinidhitva siddhantasaha vyakhya.
jarr eka-shakti gut ekhadya sanlagn prakaravar karya karat asel, tarr tyachya serve kaksha bund asatat aanee jarr too maryadit-aayami vector jaaget reshiyapane karya karat asel tarr tyala shoonya nasalela sthir vector asato. kharey tarr, nantarcha gunadharm eka-shakti gatanche vaishishtya darshavito. visheshtah, yacha arth assa aahe qee konatihi kshullak ardh-sopee pratinidhya naheet.
udaharane.
yuu "aine".
arthat, matricks sootra _ 1 chaa gut eksandh aahe. khalchya madhyavarti malikecha vapar
kuthe?
sambandhit eka-shakti gut aahet. udaharanarth, sootra _ 15 varr, madhyavarti malika hii matricks gut aahet.
ekshakti gatanchi kahee prerit udaharane dilee.
gaa "aine".
yogatmak gut sootra _ 20 haah antarbhut karoon eka ekatma gut aahe.
matricks gunakar dett aahe heey lakshat ghya
mhanoon haah eka gut embedding aahe. adhik samanyatah, nakashavarun _ 23 heey eka antarbhut sootra aahe
yojana siddhant vaparoon, sootra _ 20 heey functordware dile jaate.
kuthe?
frobeniuscha gabha.
upashreni sootra _ 29 vareel functor sootra _ 28 chaa vichar qaraa, up-karya sootra _ 30 aahe jethe
mhanoon teey frobenius endomorphism karnaldware dile jaate.
0 yaa vaishishtyanvaril ekshakti gatanche vargikaran.
0 chya vaishishtyanpeksha nilpotant laee algebranchya sandarbhat eka-shaktishali bijganitiy gatanche eka changle vargikaran aahe. lakshat thewa qee nilpotent laee bijganit heey kahee sutranche up-bijganit aahe, jase qee punaravrutti keleli sanlagn kriya akheris shoonya-nakashavar sampate. matrixchya sandarbhat, yacha arth assa aahe qee heey sootra _ 34 chey up-bijganit sootra _ 33 aahe, sootra _ 36 sathee sootra _ 35 asalele matricks.
mug, maryadit-aayami nilpotant laee algebra aanee eka-shaktishali bijganitiy gatanchya shreninchi samanata aahe.prushth 261 heey bekar-campbelle-hosdorf malika sootra _ 37 vaparoon tayar kele jaau shakate, jethe maryadit-aayami nilpotent laee bijganit dile aahe, nakasha
sootra _ 33 varr eksandh bijganitiy gut rachana dete.
dusya dishene ghatankiy nakasha konatyahi nilpotant varg matrixla eksandh matrixver gheuun jaato. shivay, jarr "yuu" haah eka parivartaneey ekshakti gut asel, tarr ghatankiy nakasha "yuu" chya khotya bijganitepasun "yuu" madhye samarupata aanato.
tippanya.
konatyahi parimanachya bijganitiyadrushtya bund kshetravaril eka-shakti gatanche tattvatah vargikaran kele jaau shakate, parantu vyavharat vargikaranachi jatilata parimananusar khoop vegane vadhate, tyamule lock pariman 6 chya aaspas kuthetari sodoon detat.
eksandh moolgami.
bijganitiy gut 'jee' chaa eka-shakti mooldrav haah 'jee' chya muldravyatil eka-shaktiman ghatakancha sanch aahe. haah 'jee' chaa eka jodalela ekshakitman samanya upasamuh aahe aanee tyat asey itar serve upasamuh aahet. jarr ekshaktiman mooldravye kshullak asateel tarr gatala reductive mhanatat. jarr "jee" haah reductive asel tarr tyacha radicals torrus aahe.
bijganitiy gatanche vighatan.
bijganitiy gut eka-shakti gut, gunak gut aanee abelian prakaranmadhye vighatit kele jaau shakatat, parantu teey kase vighatit hotat yache vidhan tyanchya aadhar kshetrachya vaishishtyanvar avalambun asate.
0 heey vaishishtyapurn.
0 chya vaishishtyanvaril, tyachya sanrachaneshi rekheey bijganitiy gatachya sanrachaneshi aanee abelian prakarashi sambandhit asalelya bijganitiy gut sootra 40 chey eka changle vighatan pramey aahe. gatancha eka sankshipt achook kram aahe prushth 8
jethe sootra _ 42 haah eka abelian prakar aahe, sootra _ 43 gunak prakarcha aahe (mhanje, sootra _ 43, bhaumitikdrushtya, sootra _ 45 chya rupatil toree aanee bijganitiy gatanche utpadan aahe) aanee sootra _ 46 haah eka eksandh gut aahe.
vaishishtyapurn "pii.
jevha aadhar kshetrache vaishishtya "p" asate tevha bijganitiy gut sutrasathi eka samaan vidhan asate _ 40: eka sarvat lahan upasamuh sootra _ 48 asey astitvat aahe qee
jorden vighatan.
eka paripoorn kshetravaril resheey bijganitiy gatacha konatahi ghatak "jee" haah ekopayogi aanee ardh-sadhya ghatakanchya "jee" yuu "aanee" jee "ess" chya pravasache utpadan "jee" = "jee" yuu "aanee" hairs "; aanee" hairs ";" jee "ess" mhanoon advitiyapane lihila jaau shakato. gut 'jee. ell.' aine. '(see) chya babatit, heey moolatah asey mhanate qee konatehi ulatya karata yenyajoge jatil matricks heey karn matricks aanee varchya trikoni matrixchya gunakarasah sanyugmit aahe, jeey (qamee-adhik) jorden-shavely vighatanchi gunakar aavrutti aahe.
gatansathi jorden vighatanchi eka aavrutti dekhil aaheh 
paripoorn kshetravaril konatahi parivartaneey rekheey bijganitiy gut haah eka-shakti gut aanee ardh-sadha gatacha gunakar asato.