
vikarnaniya matricks

raikhik beejganit mein, ack varg matricks sootra _ 1 koo vikarn yaa gair-doshpurna kahaa jaataa haye yadi yeh ack vikarn matricks key samaan haye, i.ee.yadi ack vyutkramaniya matricks sootra _ 2 aur ack vikarn matricks sootra _ 3 maujood haye tou aisaa yaa samkaksh (aisaa sootra _ 3 advitiya naheen hain.) ack parimit-ayami sadish sthaan key liye ack raikhik maanchitra sootra 5 koo vikarnaniya kahaa jaataa haye yadi sootra _ 6 kaa ack krambaddh aadhar maujood haye jisamen sootra _ 7 key igenvector shamil hain. yeye paribhaashaaen samatulya hainh yadi sootra _ 7 mein oopar key roop mein ack matricks pratinidhitva sootra _ 9 haye, tou sootra _ 2 key stambh vactor iss igenvector key sambandh mein ack aadhar banaate hain aur sootra _ 3 kii vikarn pravishtiyaan iss igenvector aadhar key sambandh mein sambandhit igenvalue hain, sootra _ 1 koo vikarnan dwara darshaaya jaataa haye, uparokt sootra _ 2 koo khojane kii prakriya haye aur 
vikarnaniya matricks aur maanchitra vishesh roop seey ganana key liye aasaan hain, ack baar jaba unake igenvalue aur igenvector gyaat hoe jaate hain. ack vikarn matricks sootra koo kewal usa shakti key liye vikarn pravishtiyon koo badhaakar ack shakti key liye badhaa sakata haye, aur ack vikarn matricks kaa nirdhaarak kewal sabhi vikarn pravishtiyon kaa utpaad haye; iss tarah kii gananaon koo aasaani seey jyaamitiya roop seey saamaanyikrit kiya jaataa haye, ack vikarn matricks ack asamaroop failaav (yaa "anisotropic scaling") haye-yeh sthaan koo maapata haye, jaisa kii ack "sajaatiya failaav" kartaa haye, lekin pratyek igenvector aksh key saath ack alag kaarak dwara, sambandhit igenvalue dwara diyaa gayaa kaarak.
ack varg matricks joe vikarn naheen haye, usey "doshpurna" kahaa jaataa haye. aisaa hoe sakata haye kii vaastavik pravishtiyon key saath ack matricks sootra _ 1 vaastavik sankhyaon parr doshpurna haye, jiska arth haye kii sootra _ 16 vaastavik pravishtiyon key saath kissy bhee vyutkram sootra _ 2 aur vikarn sootra _ 3 key liye asambhav haye, lekin yeh jatil pravishtiyon key saath sambhav haye, taaki sootra _ 1 jatil sankhyaon parr vikarn haye. udaaharan key liye, yeh ack samanya rotation matricks key liye mamla haye. 
vikarnaniya matricks key liye kaee parinam kewal ack beejganitiya roop seey bund kshetra (jaise kii jatil sankhya) parr pakad rakhate hain. iss mamle mein, vikarnaniya matricks sabhi matricks key sthaan mein ghaney hotey hain, jiska arth haye kii kissy bhee doshpurna matricks koo ack chhote seey vikshobh dwara vikarnaniya matricks mein vikrit kiya jaa sakata haye; aur jorden samanya roop pramey bataata haye kii koee bhee matricks vishisht roop seey ack vikarnaniya matricks aur ack neelpotent matricks kaa yog haye. beejganitiya roop seey bund kshetra parr, vikarnaniya matricks ardh-saral matricks key barabar hotey hain.
paribhasha.
ack varg sootra _ 20 matricks, sootra _ 1, ack kshetra sootra _ 22 mein pravishtiyon key saath vikarn yaa gair-doshpurna kahaa jaataa haye yadi ack sootra _ 20 vyutkramaniya matricks (i.ee. samanya raikhik samooh gl "n" ("f") kaa ack tatva, sootra _ 2, aisaa sootra _ 25 ack vikarn matricks haye. aupchaarik roop seey,
charitra varnan.
vikarn maanchitron aur matricks key baare mein maulik tathya nimnalikhit dwara vyakt kiya gayaa haih
nimnalikhit paryaapt (lekin aavashyak naheen) sthiti aksar upayogi hoti haye.
maan leejiye kii sootra 1 ack matricks haye yadi sootra 1 vikarnaniya haye, tou isaki koee bhee shakti haye. isake vipareet, yadi sootra _ 1 vyutkramaniya haye, tou sootra _ 22 beejganitiya roop seey bund haye, aur sootra _ 61 kuchh sootra _ 29 key liye vikarnaniya haye joe kii sootra _ 1 kii visheshta kaa ack purnank gunak naheen haye, tou yeh vikarnaniya haye. pramanah yadi sootra _ 61 vikarnaniya haye, tou sootra _ 1 koo kuchh bahupad dwara nasht curr diyaa jaataa haye jiska koee ekaadhik mool naheen haye (kyonki aur ise nyuntam bahupad dwara vibhajit kiya jaataa haye) 
jatil sankhya sootra _ 66 parr, lagbhag harr matricks vikarnaniya haye. adhik sateek roop seh jatil sootra _ 20 matricks kaa sett joe kii "vikarn naheen" haye, jise ack upasamuchchay key roop mein maanaa jaataa haye, mein labsgue maap shoonya haye. koee yeh bhee kaha sakata haye kii vikarnaniya matricks zariski topology key sambandh mein ack ghanaa upasamuchchay banaate hainh gair-vikarniya matricks visheshta bahupad key bhedbhaav key gayab honey vaale sett key ander sthit hotey hain, joe ack atee surface haye. isase ack maanak dwara dee gaee samanya ("majboot") topology mein ghanatva bhee hotaa haye. vahee baat sach naheen haye 
jorden-chevalle apghatan ack prachaalak koo usake ardh-saral (i.ee.vikarn) bhaag aur iska neelpotent bhaag. isaliye, ack matricks vikarnaniya haye yadi aur kewal tabhi jaba iska neelpotent bhaag shoonya hoe. dusare tareeke seey kahein tou, ack matricks vikarn haye yadi pratyek block key jorden roop mein koee nilpotent bhaag naheen haye; i.ee.pratyek "block" ack-ack karke ack matricks haye.
vikarnan.
yadi ack matricks sootra _ 1 koo vikarn kiya jaa sakata haye, tou vaha haye,
tabah
sootra _ 2 koo apane collum vactor sootra _ 72 key block matricks key roop mein likhana
uparokt sameekaran koo iss prakaar likhaa jaa sakata haye
tou sootra _ 2 key collum vactor sahee igenvector hain aur sambandhit vikarn pravishti sambandhit igenvalue haye. sootra _ 2 kii vyutkramta seey yeh bhee pataa chalata haye kii igenvector raikhik roop seey swatantra hain aur iska aadhar vikarnata aur vikarnan key vihit drishtikon key liye aavashyak aur paryaapt sthiti haye. sootra _ 77 key pankti vactor key baaen igenvector hain 
jaba ack jatil matricks sootra _ 78 ack hermician matricks (yaa adhik aam taur parr ack samanya matricks) haye, tou sootra _ 1 key igenvector koo ack orthonormal aadhar banaane key liye chunaa jaa sakata haye aur sootra _ 2 koo ekaatmak matricks key roop mein chunaa jaa sakata haye. yadi isake alawa, sootra _ 81 ack vaastavik samamit matricks haye, tou isake igenvector koo sootra _ 82 key orthonormal aadhar key roop mein chunaa jaa sakata haye aur sootra _ 2 koo ack orthogonal matricks key roop mein chunaa jaa sakata haye.
adhikansh vyaavhaarik kaary key liye matricks koo computer software kaa upayog karke sankhyaatmak roop seey vikarn kiya jaataa haye. ise puuraa karane key liye kaee algorithm maujood hain.
ack saath vikarnan.
matricks key ack sett koo "ack saath vikarnaniya" kahaa jaataa haye yadi ack ekal vyutkramaniya matricks sootra _ 2 maujood haye jaise kii sootra _ 25 sett mein pratyek sootra _ 1 key liye ack vikarn matricks haye. nimnalikhit pramey ack saath vikarnaniya matricks koo darshaata haih vikarnaniya matricks kaa ack sett taba aataa haye jaba sett ack saath vikarnaniya hoe.
sabhi sootra _ 20 vikarnaniya matricks kaa sett (sootra _ 88 key saath over) ack saath vikarnaniya naheen haye. udaaharan key liye, matricks
vikarnaniya hain lekin ack saath vikarnaniya naheen hain kyonki way awagaman naheen karate hain.
ack samooh mein samanya matricks kaa sanchaar hotaa haye yadi aur kewal tabhi jaba yeh ack ekaatmak matricks dwara ack saath vikarnaniya hoe; yaani, ack ekaatmak matricks sootra _ 90 maujood haye jaise kii sootra _ 91 sett mein pratyek sootra _ 1 key liye vikarn haye.
jhooth siddhant kii bhasha mein, ack saath vikarn karane yogya matricks kaa ack sett ack torrel jhooth beejganit utpann kartaa haye.
udaaharan.
matricks joe vikarn naheen hain.
samanya taur parr, ack ghoornan matricks vaastavik parr vikarn naheen hotaa haye, lekin sabhi ghoornan matricks jatil kshetra parr vikarn hotey hain. bhale hii ack matricks vikarnaniya naa hoe, lekin "sabse achchha karna" hamesha sambhav haye, aur pramukh vikarn parr igenvalue seey yukt samaan gunon key saath ack matricks dhoondhana sambhav haye, aur superdiagonal parr yaa tou ack yaa shoonya-jise jorden samanya roop key roop mein jaanaa jaataa haye.
kuchh matricks kissy bhee kshetra parr vikarn naheen hotey hain, vishesh roop seey gair-shoonya nilpotent matricks. yeh adhik aam taur parr taba hotaa haye jaba ack igenvalue kii beejganitiya aur jyaamitiya gunan ack saath naheen hoti hain. udaaharan key liye, vichaar karein
yeh matricks vikarnaniya naheen haih koee matricks sootra 90 aisaa naheen haye kii sootra 98 ack vikarn matricks haye. vaastav mein, sootra 99 mein ack igenvalue (arthaat shoonya) haye aur iss igenvalue mein beejganitiya gunan 2 aur jyaamitiya gunan 1 haye.
kuchh vaastavik matricks vaastavik parr vikarn naheen hotey hain. udaaharan key liye matricks parr vichaar karein
matricks sootra _ 101 mein koee vaastavik igenvalue naheen haye, isaliye koee vaastavik matricks sootra _ 102 naheen haye jaise kii sootra _ 103 ack vikarn matricks haye. haalaanki, humm sootra _ 101 koo vikarn curr sakate hain yadi humm jatil sankhyaon kii anumati dete hain. vaastav mein, aggar humm lete hain
phir sootra _ 103 vikarn haye. yeh pataa lagaana aasaan haye kii sootra _ 101 ack ghoornan matricks haye joe konn sootra _ 108 dwara ghadee kii vipareet disha mein ghoomata haye.
dhyaan dein kii uparokt udaharanon seey pataa chalata haye kii vikarnaniya matricks kaa yog vikarnaniya honey kii aavashyakta naheen haye.
ack matricks koo vikarn kaise kiya jaae.
ack matricks koo vikarn karna usee prakriya key roop mein haye joe isake igenvalue aur igenvector koo khojane key liye haye, iss mamle mein kii igenvector ack aadhar banaate hain. udaaharan key liye, matricks parr vichaar karein
vishisht bahupad sootra _ 110 kii jadein raikhik pranali sootra _ 111 koo hull karane vaale igenvalue hain joe igenvector sootra _ 112 detaa haye aur jabaki sootra _ 113 detaa haye kii, inn vactors key liye sootra _ 114 ack aadhar banaata haye taaki humm unhein ack parivartan-key-aadhar matricks sootra _ 2 key collum vactor key roop mein ikattha curr sakenh
sootra _ 116
humm iss sameekaran koo parivartanon key sandarbh mein dekh sakate hainh sootra _ 2 igenbasis key liye maanak aadhar letaa haye, isaliye hamaare paas haih
sootra _ 118
taaki sootra _ 119 kaa maanak aadhar isake igenvector key roop mein hoe, joe kii iska paribhaashit guna haye. 
dhyaan dein kii sootra _ 2 mein igenvectors key crum koo badalane mein igenvectors kaa koee pasandida crum naheen haye, buss vikarn roop mein igenvalue key crum koo badal detaa haye. 
matricks karyon key liye anuprayog.
vikarnan kaa upayog ack matricks kii shaktiyon kii kushalata seey ganana karane key liye kiya jaa sakata haye. 
aur baad vaale kii ganana karna aasaan haye kyonki isamen kewal ack vikarn matricks kii shaktiyaan shamil hain. udaaharan key liye, uparokt udaaharan mein humm ganana karate hain kii igenvalue sootra _ 123 key saath matricks sootra _ 1 key liyeh
iss drishtikon koo matricks ghaatiya aur anya matricks karyon key liye saamaanyikrit kiya jaa sakata haye jinhen shakti shrinkhala key roop mein paribhaashit kiya jaa sakata haye. udaaharan key liye, hamaare paas paribhaashit haih
yeh vishesh roop seey raikhik punaraavarti anukramon key liye bund roop abhivyaktiyon koo khojane mein upayogi haye, jaise kii fibonacci sankhya.
vishesh anuprayog.
udaaharan key liye, nimnalikhit matricks parr vichaar keejiyeh
sootra _ 127 kii vibhinn shaktiyon kii ganana karane seey ack aashcharyajanak pattern kaa pataa chalata haih
uparokt ghatana koo ise puuraa karane key liye vikarn karke samajhaaya jaa sakata haye, hamein sootra 129 key aadhar kii aavashyakta haye jisamen ack aise igenvector aadhar key igenvector shamil hain joe dwara diyaa gayaa haye
jahan ee "aaee" aar "aine" key maanak aadhar koo darshaata haye. aadhar kaa vipareet parivartan dwara diyaa jaataa haye
seedhi ganana seey pataa chalata haye kii
iss prakaar, "a" aur "b" kramshah u aur v key anuroop igenvalue hain. matricks gunan kii raikhikta seey, hamaare paas haye kii
maanak aadhar parr vaapas jaanaa, hamaare paas haye
matricks key roop mein vyakt kiye gaye poorvavarti sambandh hain -
iss prakaar uparokt ghatana kii vyakhya kii jaati haye.
quantum yaantrik anuprayog.
quantum yaantrik aur quantum rasayanik gananaon mein matricks vikarnan sabse adhik baar lagu kii jaane vaali sankhyaatmak prakriyaon mein seey ack haye. iska mool kaaran yeh haye kii samay-swatantra schroedinger sameekaran ack igenvalue sameekaran haye, halanki ack anant ayami sthaan (ack hilbert sthaan) parr adhikansh bhautik sthitiyon mein.
ack bahut hii samanya sanniktan hilbert sthaan koo seemit ayam mein kaatna haye, jisake baad schroedinger sameekaran koo ack vaastavik samamit, yaa jatil hermician matricks kii ack igenvalue samasya key roop mein taiyar kiya jaa sakata haye. aupchaarik roop seey yeh anuman parivartansheel siddhant parr adharit haye, joe neeche seey bandhe hamilton key logon key liye maanya haye.
pratham-crum vikriti siddhant bhee apakshayi avasthaon key liye matricks igenvalue samasya kii or leey jaataa haye.

