
sahayak matricks

raikhik beejganit mein, ack varg matricks kaa sahayak yaa shaastriya sahayak isake saha-kaarak matricks kaa sthanantaran haye aur ise isake dwara darshaaya jaataa haye. ise kabhi-kabhi sahayak matricks yaa "sahayak" key roop mein bhee jaanaa jaataa haye, halanki aaj kaa baad vaalaa shabd aam taur parr ack alag avadharana koo sandarbhit kartaa haye, sahayak prachaalak joe ack matricks key liye sanyugmit transpose haye.
ack matricks kaa utpaad isake sahayak key saath ack vikarn matricks detaa haye (pravishtiyaan joe mukhya vikarn parr naheen hain, shoonya hain) jinaki vikarn pravishtiyaan mool matricks key nirdhaarak hainh
jahan samaan aakar kaa pehchaan matricks haye. natijatan, ack vyutkramaniya matricks kaa gunak vyutkram isake sahayak koo isake nirdhaarak seey vibhajit karke paaya jaa sakata haye.
paribhasha.
kaa sahayak saha-kaarak matricks kaa sthanantaran haye,
adhik vistaar seey, maan leejiye kii yeh ack ekaatmak commutative ring haye aur ack matricks haye jisamen seey pravishtiyaan hain. - kaa "minor", niroopit, matricks kaa nirdhaarak haye joe pankti aur collum koo hataane key parinamswarup hotaa haye. saha-kaarak matricks vaha matricks haye jiska pravesh "saha-kaarak" haye, joe-minor gunaa ack sanket kaarak haih
iska sahayak transpose haye, yaani vaha matricks jiska pravesh saha-kaarak haye,
mahatvapurna parinam.
sahayak koo iss tarah paribhaashit kiya gayaa haye kii isake sahayak key saath utpaad ack vikarn matricks detaa haye jisaki vikarn pravishtiyaan nirdhaarak hoti hain. vaha haye,
pehchaan matricks kahaan haye. yeh nirdhaarak key lapless vistaar kaa parinam haye.
uparokt sootra kaa tatparya matricks beejganit mein maulik parinamon mein seey ack haye, joe kii vyutkramaniya haye yadi aur kewal tabhi jaba iska vyutkramaniya tatva hoe. jaba yeh hotaa haye, tou uparokt sameekaran seey laabh hotaa haye
udaaharan.
1 × 1 samanya matricks.
choonki 0 x 0 matricks kaa nirdhaarak 1 haye, isaliye kissy bhee 1 × 1 matricks (jatil scalar) kaa sahayak sootra _ 7 haye. usa sootra kaa avalokan karein _ 8
2 × 2 samanya matricks.
2 × 2 matricks kaa sahayak
haye
pratyaksh ganana dwara,
iss mamle mein, yeh bhee sach haye kii ((aeye)) = (aeye) aur isaliye (aeye) = aeye.
3 × 3 samanya matricks.
3 × 3 matricks parr vichaar karein
iska saha-kaarak matricks haye
kahaan
iska sahayak isake saha-kaarak matricks kaa sthanantaran haye,
3 × 3 sankhyaatmak matricks.
ack vishisht udaaharan key roop mein, hamaare paas haye
yeh jaanchana aasaan haye kii sahayak vyutkram samay nirdhaarak haye.
doosari pankti mein, sahayak key teesare stambh kii ganana nimnanusar kii gaee thee. sahayak kaa (2,3) pravesh ack kaa (3,2) saha-kaarak haye. iss saha-kaarak kii ganana mool matricks key teesari pankti aur dusare collum koo hataakar praapt upa-matricks kaa upayog karke kii jaati haye.
(3,2) saha-kaarak iss upa-matricks key nirdhaarak kaa ack sanket gunaa haih
aur yeh sahayak kaa (2,3) pravesh haye.
guna.
kissy bhee matricks key liye, praathamik ganana seey pataa chalata haye kii sahayakon mein nimnalikhit guna hotey hainh
jatil sankhyaon parr,
maan leejiye kii yeh ack aur matricks haye. taba
ise teen tarikon seey saabit kiya jaa sakata haye. ack tarika, joe kissy bhee commutative ring key liye maanya haye, vaha haye kauchi-binet sootra kaa upayog karke ack pratyaksh ganana. doosara tarika, vaastavik yaa jatil sankhyaon key liye maanya haye, pahale yeh avalokan karna haye kii vyutkramaniya matricks key liye aur,
kyonki pratyek gair-vyutkramaniya matricks vyutkramaniya matricks kii seema haye, sahayak kii nirantarta kaa tatparya taba hotaa haye kii sootra taba sahee rahata haye jaba ack vyutkramaniya hotaa haye yaa naheen hotaa haye.
pichhale sootra kaa ack parinam yeh haye kii, kissy bhee gair-nakaratmak purnank key liye,
yadi vyutkramaniya haye, tou uparokt sootra bhee nakaratmak key liye haye.
pehchaan seey
humm anuman lagaate hain
maan leejiye kii yeh saath mein aataa haye. baaen aur daaen parr pehchaan koo gunaa karna saabit kartaa haye kii
yadi vyutkramaniya haye, tou iska tatparya yeh haye kii yeh bhee isake saath aataa haye. vaastavik yaa jatil sankhyaon parr, nirantarta kaa tatparya haye kii jaba vyutkramaniya naheen hoe taba bhee saath mein yatra hoti haye.
anta mein, dusare pramaan kii tulanaa mein ack adhik samanya pramaan haye, jisake liye kewal yeh aavashyak haye kii ack "n" × "n" matricks mein cumm seey cumm 2 "n" + 1 tatvon (e.) key saath ack kshetra parr pravishtiyaan houn.jee. purnank modulo 11 parr 5 × 5 matricks). t" mein ack bahupad haye jisaki adhikatam "n" degree haye, isaliye isaki adhikatam "n" jadein hain. dhyaan dein kii "ij" & hairsp; th pravishti adhikatam crum "n" kii ack bahupad haye, aur isi tarah key liye. ij" aur "hairsp"; th pravishti parr yeye dou bahupad cumm seey cumm "n" + 1 ankon parr sahmat hain, kyonki hamaare paas cumm seey cumm "n" + 1 tatva hain jahaan vyutkramaniya haye, aur hamane vyutkramaniya matricks key liye pehchaan saabit kii haye. degree "aine" kii bahupad joe "aine" + 1 ankon parr sahmat hoti hain, samaan honey chaahiye (unhein ack dusare seey ghataakar aur aapke paas adhikatam "aine" degree key bahupad key liye "aine" + 1 mool hain-ack virodhabhas jaba taka kii unakaa antar samaan roop seey shoonya naa hoe). choonki dou bahupad samaan hain, way "t" key pratyek moolya key liye samaan moolya lete hain. iss prakaar, way samaan moolya lete hain jaba "t" = 0 hotaa haye. 
uparokt gunon aur anya praathamik gananaon kaa upayog karate hue, yeh dikhaana seedha haye kii yadi nimnalikhit gunon mein seey ack haye, tou yeh bhee kartaa haih
yadi vyutkramaniya haye, tou, jaisa kii oopar ullekh kiya gayaa haye, nirdhaarak aur vyutkram key sandarbh mein ack sootra haye. jaba vyutkramaniya naheen hotaa haye, tou sahayak vibhinn lekin nikatata seey sambandhit sootron koo santusht kartaa haye.
collum pratisthaapan aur crammer kaa niyam.
stambh vactors mein vibhaajanah
maan leejiye kii yeh aakar kaa ack collum vector haye. stambh koo iss prakaar seey badalkar banaae gaye matricks koo thheek karein aur usa parr vichaar karein -
lapless iss matricks key nirdhaarak koo collum key saath vistaarit kartaa haye. parinam utpaad kii pravishti haye. vibhinn sambhaavit ghatakon key liye inn nirdharakon koo ekatra karane seey stambh vactors kii samaantaa milati haye.
iss sootra kaa nimnalikhit thhos parinam haye. samikaranon kii raikhik pranali parr vichaar karein
maan leejiye kii yeh gair-ekal haye. iss pranali koo baain or seey gunaa karna aur nirdhaarak upaj seey vibhajit karna
iss sthiti mein pichhale sootra koo lagu karane seey crammer kaa niyam milata haye,
kahaan kii pravishti haye.
vishisht bahupad.
visheshta bahupad koo honey dein
pahala vibhajit antar degree kaa ack samamit bahupad haye,
isake sahayak seey gunaa karein. kyonki kellie-hamilton pramey seey kuchh praathamik herfer prakat hotey hain
vishesh roop seey, key sankalp koo paribhaashit kiya gayaa haye
aur uparokt sootra key anusaar, yeh isake barabar haye
jakobi kaa sootra.
nirdhaarak key vyutpann key liye jakobi key sootra mein bhee sahayak dikhaai detaa haye. yadi lagaatar bhinn haye, tou
yeh iss prakaar haye kii nirdhaarak kaa kull vyutpann sahayak kaa sthanantaran haih
kellie-hamilton sootra.
maan leejiye kii yeh visheshta bahupad haye. kellie-hamilton pramey bataata haye kii
sthir pada koo alag karane aur sameekaran koo gunaa karane seey sahayak key liye ack abhivyakti milati haye joe kewal isake gunank parr nirbhar karati haye. inn gunankon koo poorna ghaatiya ghanti bahupadon kaa upayog karane kii shaktiyon key nishan key sandarbh mein spasht roop seey darshaaya jaa sakata haye. parinami sootra haye
jahan kaa ayam haye, aur yog koo liya jaataa haye aur raikhik diophantine sameekaran koo santusht karane key sabhi anukram
2 × 2 mamle key liye, yeh detaa haye
3 × 3 mamle key liye, yeh detaa haye
4 × 4 mamle key liye, yeh detaa haye
vahee sootra seedhe fedive-leverior algorithm key samaapt karane vaale charan kaa anusaran kartaa haye, joe kushalata seey visheshta bahupad koo nirdhaarit kartaa haye.
baahari algebra seey sambandh.
sahayak koo baahari algebra kaa upayog karke amoort shabdon mein dekhaa jaa sakata haye. ack-ayami sadish sthaan honey dein. baahari utpaad ack dwilingi jodee koo paribhaashit kartaa haye
amoort roop seey, sootra _ 57 samroopata haye, aur iss tarah key kissy bhee samroopata key tahat baahari utpaad ack sahee jodee haye. isaliye, yeh ack samroopata utpann kartaa haye
spasht roop seey, yeh jodee sootra _ 59 parr bhejati haye, jahan
maan leejiye kii yeh ack raikhik parivartan haye. aiss. tii. kii baahari shakti dwara peechhe hatane seey rikt sthaan kaa rupankan hotaa haye. iska sanyojan haye
yadi isake vihit aadhar key saath sampann haye, aur yadi iss aadhar mein matricks haye, tou iska sahayak haye. yeh dekhane key liye kii kyon, sootra _ 62 aadhar dein
ack aadhar sadish koo thheek karein. sootra _ 64 key tahat chhavi iss baat seey nirdhaarit hoti haye kii yeh kahaan aadhar vactor bhejata haih
vactor key aadhar parr, kii aiss. tii. baahari shakti haye
inn shabdon mein seey pratyek sootra _ 67 key tahat shoonya taka maanchitrit hotaa haye sivaay shabd key. isaliye, sootra _ 67 kaa pullback raikhik parivartan haye jisake liye
vaha haye, yeh barabar haye
sootra _ 64 key vyutkram koo lagu karane seey pataa chalata haye kii iska sahayak raikhik parivartan haye jisake liye
natijatan, iska matricks pratinidhitva iska sahayak haye.
yadi ack aantarik utpaad aur ack aayatan roop key saath sampann haye, tou maanchitra koo aage vighatit kiya jaa sakata haye. iss mamle mein, hodge starr operator aur doharikaran key sanyojan key roop mein samajha jaa sakata haye. vishesh roop seey, yadi aayatan roop haye, tou yeh aantarik utpaad key saath milkar ack samroopata nirdhaarit kartaa haye.
yeh ack samroopata koo prerit kartaa haye
ack sadish raikhik karyatmak seey mell khaata haye
hodge starr operator kii paribhasha key anusaar, yeh raikhik karyatmak dohare seey haye. vaha haye, barabar.
uchch sahayak.
ack matricks honey dein, aur thheek karein. iska oonchaa sanyojan ack sootra _ 76 matricks haye, jise darshaaya gayaa haye, jisaki pravishtiyaan aakar upasamuchchay aur key dwara anukramit kii gaee hain. maan leejiye aur kramshah aur key poorak koo darshaaie. sootra _ 77 koo una panktiyon aur stambhon koo rakhane key upmatrics koo bhee darshaata haye jinake soochkaank kramshah aur mein hain. taba pravesh haye
jahan aur kramshah tatvon kaa yog haye.
uchch sahayakon key buniyaadi gunon mein shamil hainh
uchch sahayakon koo amoort beejganitiya shabdon mein samanya sahayak key samaan roop seey paribhaashit kiya jaa sakata haye, sootra _ 82 aur sootra _ 83 key liye sootra _ 80 aur sootra _ 81 koo pratisthaapit karate hue.
punaraavritt sahayak.
ack baar ack vyutkramaniya matricks key sahayak koo lete hue ack baar upaj
udaaharan key liye,

