
hermician matricks

ganitat, hermician matricks (kinwa self-adjoint matricks) haah eka jatil varg matricks aahe joou tyachya swatachya sanyugma transposechya barobaricha aahe-mhanje,-vya panktitil ghatak aanee-vya stambhatil ghatak haah ghatak-vya panktitil aanee-vya stambhatil, serve nirdeshankansathi aanih
kinwa matricks swarupatah
sootra _ 1
hermician matricks heey vastavik samamiteey matrixche jatil vistar mhanoon samajale jaau shakatat.
jarr matricks sootra _ 2 chey sanyugmit transpose sootra _ 3 dware darshavile gele tarr hermician gunadharm sankshiptapane asey lihile jaau shakate
hermician matricksna charles hermiteche naav denyat aale aahe, jyanni 1855 madhye heey dakhavun dile qee yaa swarupachya matrixmadhye nehmi vastavik eigenvalues asalelya vastavik samamiteey matricks gunadharm samayik keela jaato. samanya vaparatil itar samatulya sanketank sootra 4 aahet, jaree quantum yantrikimadhye sootra 5 chaa arth samanyatah keval jatil sanyugma aahe, sanyugma transpose nahee.
paryayi vaishishtye.
hermician matricks anek samatulya marganni darshavile jaau shakatat, jyapaiki kahee khalee soochibaddha aaheth
bajushi samanata.
jarr aanee keval jarr teey tyachya sanlagn bhagachya barobariche asel, mhanje teey samadhankarak asel, tarr varg matricks sootra _ 2 heey hermician aahe.
sootra _ 7
vector sootra _ 8 chya konatyahi jodisathi jethe sootra _ 9 antargat utpadan karya darshavate.
self-adjoint operatorchi adhik samanya sankalpana paribhashit karnyacha dekhil haah marg aahe.
dvighat swarupanche vastav.
sootra _ 10 matricks sootra _ 2 heey hermician aahe jarr aanee fakt jarr
sootra _ 12
varnakrameey gunadharm.
jarr aanee keval jarr teey vastavik igenvalues ekeri karnarekheey asel tarr varg matricks sootra _ 2 heey hermician aahe.
anuprayog.
hermician matricks heey quantum yantrikisathi moolbhut aahet kaaran teey aavashyakpane vastavik eigenvalue asalelya operatorche varnan karatat. kahee quantum sthitee sutravar operator sootra 15 chey eigenvalue sootra _ 14 heey operatorchya sambhavya mojamapachya parinamanpaiki eka aahe, jyamule vastavik eigenvalue asalelya operatorchi aavashyakta aavashyak aahe.
udaharane aanee upay.
yaa vibhagat, matricks sootra _ 17 chey sanyugmit transpose sootra _ 18 mhanoon darshavile jaate, matricks sootra _ 17 chey transpose sootra _ 20 mhanoon darshavile jaate aanee matricks sootra _ 17 chey sanyugmit sootra _ 22 mhanoon darshavile jaate.
khaleel udaharan pahah
sootra _ 23
karn ghatak vastavik asale pahijet, kaaran teey tyanche swatache jatil sanyugma asane aavashyak aahe.
hermician matrixchya suprasiddha kutumbanmadhye pauli matricks, jail-mann matricks aanee tyanche samanyikaran yancha samavesh hoto. saiddhantik bhautikshastrat asha hermician matricksna anekada kalpanik gunankanni gunale jaate, jyacha parinam tirkas-hermician matrixmadhye hoto.
yethe, aapan eka amoort udaharan vaparoon aanakhi eka upayukt hermician matricks deu karato. jarr chauras matricks sootra _ 17 heey tyachya sanyugmit transposes matrixchya gunakarachya barobariche asel, mhanje sootra _ 25 tarr sootra _ 17 heey eka hermician positive ardh-nishchit matricks aahe. yavyatirikt, jarr sootra _ 27 heey pankti purn shreniche asel, tarr sootra _ 17 heey sakaratmak nishchit aahe.
malmatta.
mukhya karn moolye vastavik aahet.
konatyahi hermician matrixchya mukhya karnavaril (varr davikadoon khaloon ujavikade) nondi vastavik asatat.
keval mukhya karn pravishtya aavashyakpane vastavik asatat; joparyant tirkas-viruddha nondi jatil sanyugma asatat, toparyant hermician matrixmadhye tyanchya tiryak ghatakanmadhye aniyantrit jatil-moolya asalelya nondi asoo shakatat.
samamiteey.
keval vastavik nondi asalela matricks jarr aanee keval jarr too hermician matricks asel tarach samamiteey asato. vastavik aanee samamiteey matricks heey fakt hermician matrixche eka vishesh prakaran aahe.
tarr, jarr vastavik samamiteey-virodhi matrixla kalpanik ekak sootra _ 29 chya vastavik gunakarane gunale tarr teey hermician banate.
samanya.
pratyek hermician matricks haah eka samanya matricks aahe. mhanje sootra _ 30
vikarn karnyayogya.
maryadit-aayami varnakrameey pramey asey mhanate qee konatyahi hermician matrixche ekatmak matrixdware vikarnakaran kele jaau shakate aanee parinami vikarn matrixmadhye keval vastavik nondi aahet. heey soochit karate qee pariman asalelya hermician matrixchi serve eigenvalue vastavik aahet aanee tyat resheey swatantra eigenwector aahet. shivay, hermician matrixmadhye bhinn eigenvaluesathi orthogonal igenvectors asatat. jaree degenerate eigenvalues asali taree, igenvectors samavesh asalela orthogonal aadhar shodhane nehmich shakya aahe.
hermician matrixchi berries.
konatyahi done hermician matrixchi berries hermician aahe.
yachya ulat hermician aahe.
ulat karata yenyajogya hermician matrixcha ulat dekhil hermician asato.
hermician matrixche sahyogi utpadan.
done hermician matrixche gunakar aanee jarr aanee fakt jarr tarr hermician aahe.
 = \Overline {b ^ \mathsf {t} a ^ \mathsf {t}} = \Overline {b ^ \mathsf {t}}\\Overline {a ^ \mathsf {t}} = b ^ \mathsf {h} a ^ \mathsf {h} = b.</ganeet> asey sootra _ 31 jarr aanee sootra _ 32 asel tarach
mhanoon jarr hermician asel aanee purnank asel tarr hermician aahe.
"abah" hermician.
jarr 'aaa' aanee 'buub' heey hermician asateel tarr 'abah' dekhil hermician aahe.
jatilatesathi teey kharey aahe.
aniyantrit jatil moolya asalelya vectorsathi utpadan sootra _ 33 heey vastavik aahe kaaran sootra _ 34 heey quantum bhautikshastrat visheshtah mahatvache aahe jethe hermician matricks heey operator aahet jeey pranaliche gunadharm mojatat, e.jee. ekoon firki, jee vastavik asali pahije.
jatil hermician heey vector space tayar karate.
hermician complexes-buy-matricks heey complexes sankhyanvar vector space tayar karat naheet, kaaran identity matricks hermician aahe, parantu nahee. tathapi, jatil hermician matricks 'duu' vastavik sankhyanvar vector space tayar karatat. jatil matrixchya-aayami vector spacemadhye, jatil hermician matricks parimananchi upasthan tayar karatat. jarr-buy-matrixla aaa sthiteet aanee itaratra shoonyasah soochit kele tarr eka aadhar (frobanius aateel utpadanachya sandarbhat orthonormal) chey varnan khalilpramane kele jaau shakateh
sootra _ 35
formchya matrixchya sanchasah
sootra _ 36
aanee matricks
sootra _ 37
jethe sootra _ 38 heey kalpanik ekak darshavate, sootra _ 39
eka udaharan asey aahe qee chaar pauli matricks serve jatil 2-buy-2 hermician matrixchya vector spesasathi sampoorn aadhar tayar karatat.
igendecomposition.
jarr hermician matrixche orthonormal igenvectors sootra _ 40 nivadle gele aanee matrixche stambh mhanoon lihile gele, tarr eka igendecomposition sootra _ 41 aahe jethe sootra _ 42 aanee mhanoon 
sootra _ 43
jethe sootra _ 44 heey karn matricks sootra _ 45 chya karnavaril eigenvalue aahet.
ekvachan moolye.
sootra _ 2 chee ekvachan moolye hii tyachya eigen mulyanchi paripoorn moolye aaheth
sootra _ 2 madhye igendecomposition sootra _ 48 asalyane, jethe sootra _ 49 heey eka ekatmak matricks aahe (tyache stambh orthonormal vector aahet; varr pahaa), sootra _ 2 chey ekvachan moolya vighatan sootra _ 51 aahe, jethe sootra _ 52 aanee sootra _ 53 heey karn matricks aahet jyat anukrame nirpeksh moolye sootra _ 54 aanee chinhanche sootra _ 55 aahet. sootra _ 57 ekatmak aahe, kaaran sootra _ 58 chey stambh fakt sootra _ 59 neey gunale jaat aahet. sootra _ 52 madhye sootra _ 2 chee ekvachan moolye aahet, mhanje, tyachya eigen mulyanchi paripoorn moolye.
vastavik nirdharak.
hermician matrixcha nirdharak vastavik aaheh
hermician aanee sque-hermician matrixmadhye vighatan.
hermician matrixshee sambandhit atirikt tathyanmadhye khaleel goshtincha samavesh hotoh
rele bhagank.
ganitat, dilelya jatil hermician matricks aanee shunyetar vectorsathi, rele bhagakar sootra _ 65 ashee paribhashit kelee jaateh
sootra _ 66
vastavik matricks aanee vectorsathi, hermician asanyachi sthitee samamiteey asanyaparyant qamee hote aanee konatyahi shoonya nasalelya vastavik scalar sutrasathi sanyugmit transpose sootra _ 67 nehamichya transpose sootra _ 68 sootra _ 69 madhye dekhil, lakshat thewa qee hermician (kinwa vastavik samamiteey) matrixmadhye vastavik eigenvalues asatat.
heey darshavile jaau shakate qee, dilelya matrixsathi, rele bhagakar tyachya kiman moolya sootra _ 71 (m chey sarvat lahan eigenvalue) paryant pohochato jevha sootra _ 72 heey sootra _ 73 (sambandhit eigenwector) asate. tyachpramane, sootra _ 74 aanee sootra _ 75
serve aaygan mulyanchi achook moolye midvanyasathi kiman-kamal prameyamadhye rele bhagacha vapar keela jaato. igenvactor andajatoon eigenvalue andaj prapt karnyasathi eigenvalue algoridammadhe dekhil yacha vapar keela jaato. visheshtah, heey rele bhagank punaravruttisathi aadhar aahe.
rele bhagachya shrenila (matrixsathi jeey aavashyakpane hermician nahee) sankhyatmak shreni (kinwa karyatmak vishleshanat spectrum) mhanatat. jevha matricks hermician asato, tevha sankhyatmak shreni varnakrameey manakachya barobarichi asate. ajunahi karyatmak vishleshanat, sootra _ 76 laa varnakrameey trijya mhanoon olakhale jaate. see *-algebra kinwa bijganitiy quantum yantrikichya sandarbhat, nishchit aanee bijganitachya madhyamatoon badalat asalelya rele bhagankala sanlagn karnare karya bijganitachi 'vector sthitee' mhanoon sandarbhit kele jaeel.