matricks vibhajan

sankhyaatmak raikhik beejganit key ganitiya vishay mein, ack matricks vibhajan ack abhivyakti haye joe ack diye gaye matricks koo matricks key yog yaa antar key roop mein darshaati haye. kaee punaraavritt vidhiyaan (udaaharan key liye, vibhedak samikaranon kii pranaliyon key liye) tribhujakar matricks kii tulanaa mein adhik samanya matricks seey judey matricks samikaranon key pratyaksh samadhan parr nirbhar karati hain. inn matricks samikaranon koo aksar seedhe aur kushalata seey hull kiya jaa sakata haye jaba ack matricks vibhajan key roop mein likhaa jaataa haye. takneek koo richard aiss dwara taiyar kiya gayaa thaa. 1960 mein varga.
niyamit vibhajan.
humm matricks sameekaran koo hull karna chaahte hain
jahan a ack diyaa gayaa "n" × "n" gair-ekal matricks haye, aur k "n" ghatakon key saath ack diyaa gayaa collum vector haye. humm matricks koo aeye mein vibhajit karate hain
jahan b aur c "n" × "n" matricks hain. yadi, ack manmaane "n" × "n" matricks m key liye, m mein gair-nakaratmak pravishtiyaan hain, tou humm m ≤ 0 likhate hain. yadi m mein kewal sakaratmak pravishtiyaan hain, tou humm m> 0 likhate hain. isi tarah, yadi matricks m1 − m2 mein gair-nakaratmak pravishtiyaan hain, tou humm m1 ≤ m2 likhate hain.
paribhaashaah a = b − c, a kaa niyamit vibhajan haye yadi b − 1 ≤ 0 aur c ≤ 0 haye.
humm maan lete hain kii roop key matricks sameekaran
jahan g ack diyaa gayaa collum vector haye, usey seedhe vector x key liye hull kiya jaa sakata haye. yadi () aeye key niyamit vibhajan kaa pratinidhitva kartaa haye, tou punaravritti vidhi
jahan x (0) ack manmana sadish haye, vahaan kiya jaa sakata haye. samatulya roop mein, humm () koo roop mein likhate hain.
matricks d = b − 1c mein gair-nakaratmak pravishtiyaan hain yadi () a key niyamit vibhajan kaa pratinidhitva kartaa haye.
yeh dikhaaya jaa sakata haye kii yadi a −1> 0 haye, tou sootra _ 1 <1, jahaan sootra _ 1 d kii varnakramiya trijya kaa pratinidhitva kartaa haye, aur iss prakaar d ack abhisari matricks haye. parinamswarup, punaravritti vidhi () anivarya roop seey abhisari haye.
yadi isake alawa, vibhajan () koo chunaa jaataa haye taaki matricks b ack vikarn matricks hoe (vikarn pravishtiyon key saath sabhi gair-shoonya, kyonki b koo vyutkramaniya honah chaahiye), tou b koo raikhik samay mein ulta kiya jaa sakata haye (samay jatilata dekhein).
matricks punaravritti vidhiyaan.
kaee punaravritti vidhiyon koo ack matricks vibhajan key roop mein varnit kiya jaa sakata haye. yadi matricks a kii vikarn pravishtiyaan sabhi gair-shoonya hain, aur humm matricks a koo matricks yog key roop mein vyakt karate hain
jahan d, a kaa vikarn bhaag haye, aur u aur l kramshah sakhti seey oopari aur nichale trikoniya "n" × "n" matricks hain, tou hamaare paas nimnalikhit hain.
jakobi vidhi koo vibhajan key roop mein matricks key roop mein darshaaya jaa sakata haye.
goss-seedle vidhi koo vibhajan key roop mein matricks key roop mein darshaaya jaa sakata haye.
kramik atee-shithilata kii vidhi koo vibhajan key roop mein matricks key roop mein darshaaya jaa sakata haye.
udaaharan key liye.
niyamit vibhajan.
sameekaran () mein, chalo
aaie humm vibhajan () koo lagu karein joe jacoby vidhi mein upayog kiya jaataa haih humm a koo iss tarah seey vibhajit karate hain kii b mein a key vikarn tatvon key "sabhi" hotey hain, aur c mein a key upp-vikarn tatvon key "sabhi" hotey hain, joe nakaratmak hotey hain. (nishchit roop seey yeh ack matricks koo dou matricks mein vibhajit karane kaa ekmatra upayogi tarika naheen haye.) hamaare paas haye
choonki b − 1 ≤ 0 aur c ≤ 0, vibhajan () ack niyamit vibhajan haye. choonki a − 1> 0, varnakramiya trijya sootra _ 1 <1. (d key anumaanit igenvalue sootra _ 6 hain) isaliye, matricks d abhisari haye aur vidhi () aavashyak roop seey samasya () key liye abhisari haye. dhyaan dein kii a key vikarn tatva sabhi shoonya seey adhik hain, a key upp-vikarn tatva sabhi shoonya seey cumm hain aur a sakhti seey vikarn roop seey pramukh haye.
vidhi () samasya parr lagu hoti haye () phir roop letee haye
sameekaran () kaa sateek samadhan haye
sameekaran () key liye pahale kuchh punaraavrittiyaan neeche dee gaee talika mein soochibaddh hain, joe shuru mein hain. talika seey koee bhee dekh sakata haye kii vidhi spasht roop seey samadhan () mein parivartit hoe rahee haye, halanki dheere-dheere.
jakobi vidhi.
jaisa kii oopar kahaa gayaa haye, jacoby vidhi () oopar pradarshit vishisht niyamit vibhajan () key samaan haye.
goss-seedle vidhi.
choonki samasya () mein matricks kii vikarn pravishtiyaan sabhi gair-shoonya hain, isaliye humm matricks koo vibhajan () key roop mein vyakt curr sakate hain, jahan
taba humm
samasya parr lagu goss-seedle vidhi () roop letee haye
sameekaran () key liye pahale kuchh punaraavrittiyaan neeche dee gaee talika mein soochibaddh hain, joe shuru mein hain. talika seey koee bhee dekh sakata haye kii vidhi spasht roop seey samadhan () mein parivartit hoe rahee haye, joe oopar varnit jacoby vidhi kii tulanaa mein kuchh adhik tej haye.
kramik atee-vishram vidhi.
maan leejiye "ω" = 1.1. lagaatar adhik-shithilata vidhi key liye samasya () mein matricks key vibhajan () kaa upayog karate hue, hamaare paas haye
samasya parr lagu kramik atee-vishram vidhi () roop letee haye.
sameekaran () key liye pahale kuchh punaraavrittiyaan neeche dee gaee talika mein soochibaddh hain, joe shuru mein hain. talika seey koee bhee dekh sakata haye kii vidhi spasht roop seey samadhan () mein parivartit hoe rahee haye, joe oopar varnit goss-seedle vidhi kii tulanaa mein thodi tej haye.

