
spectral sidhaantham

ganithashaasthrathil, prathyekichu leanier algibrayilum functional analysisilum, oru leanier oopparettareyo matrixineo diagonalise cheyyaan kazhiyumbozhulla falamaanu spectral sidhaantham (athaayath, chila adisthaanathil diagonal matrixaayi prathinidheekarikkunnu). diagonalise cheyyaavunna matricks ulppedunna kanakkukoottalukal palappozhum anubandha diagonal matricks ulppedunna valare lalithamaaya kanakkukoottalukalaayi kuraykkaan kazhiyumennathinaal ithu valare upayogapradamaanu. diagonalisation enna ashayam parimitha-dimensional vekttar speysukalile oopparettarmaarkku thaarathamyena nereyaanengilum anantha-dimensional speysukalile oopparettarmaarkku kurachu parishkkaranangal aavasyamaanu. pothuve, spectral sidhaantham gunikkal oopparettarmaarkku maathrkayaakkaan kazhiyunna oru tharam leanier oopparettarmaare thirichariyunnu, avaa oraalkk kandethaan kazhiyunnathra lalithamaanu. kooduthal amoorthamaaya bhashayil, spectral sidhaantham commutative sii *-aaljibrakalekkurichulla oru prasthaavanayaan. charitraparamaaya oru veekshanathinaayi spectral sidhaanthavum kaanuka.
spectral sidhaantham baadakamaakunna oopparettarmaarude udaaharanagal svayam-adjayint oopparettarmaaraanu allengil hilbert speysukalile saadhaarana oopparettarmaaraanu.
spectral sidhaantham oopparettar pravartthikkunna adisthaana vekttar spacinte spectral decomposition ennariyappedunna oru canonical decomposition nalkunnu.
augustin-luis kauchi cemetric maatriksukalude spectral sidhaantham theliyichu, i.iii.oro yadhaartha, samamiti matrix diagonalise cheyyaavunnavayaanu. koodaathe, nirnnaayakangalekkurich chittayaya aadyathethu kauchiyaanu. joonn voonn neuman saamaanyavalkkaricha spectral sidhaantham innu orupakshe oopparettar sidhaanthathinte ettavum pradhanappetta falamaanu.
eee lekhanam pradhaanamaayum shradha kendreekarikkunnathu ettavum lalithamaaya spectral sidhaanthathilaanu, athu oru hilbert speysile oru svayam-adjayint oopparettarkku. ennirunnaalum, mukalil soochippichathupole, spectral sidhaantham oru hilbert speysile saadhaarana oopparettarmaarkkum baadhakamaan.
parimitha-dimensional cases.
hermissian maappukalum hermissian maatriksukalum.
formula _ 1-ll oru hermissian matricks pariganichukondu nammal aarambikkunnu (ennaal inipparayunna charcha formula _ 2-leeii cemetric maatriksukalude kooduthal niyantritha kesumaayi poruthappedum). positteev dephinit sesquilinear innar prodakt formula _ 3 ulla oru parimitha-dimensional complexes innar prodakt spacil oru hermissian mapp njangal pariganikkunnu. formula _ 4 leeii hermissian avastha ardhamaakkunnathu ellaavarkkum,
oru thulyamaaya vyavastha, evideyaanu hermissian konjugettu. oru hermissian matricks upayogichu thirichariyunna saahacharyathil, athinte matricks athinte konjugettu transpose upayogichu thirichariyaan kazhiyum. (oru yadhaartha matricks aanenkil, ithu oru cemetric matrixinu thulyamaanu.)
eee vyavastha soochippikkunnathu oru hermissian bhoopadatthinte ellah eejan moolyangalum yadhaarthamaanennu oru eejanvektar aayirikkumbol athu kesil prayogikkaan ithu mathiyakum. (oru leanier maappinte oru eejanvektar chila scalerukalkku (poojyamallaatha) vector aanennu oorkkuka. moolyam anubandha eejan moolyamaanu. koodaathe, aijan moolyangal swabhaavasavisheshathayulla polynomialinte verukalaanu.)
sidhaantham. hermissian oonn aanenkil, athinte egenvectorukal ulkkollunna oru oorthoonoormal adisthaanam nilavilundu. oro eejan moolyavum yadhaarthamaanu.
skailerukalude adisthaana feeld sankeernna samkhyakalaaya kesinte thelivu njangal nalkunnu.
beejaganithatthinte adisthaana sidhaantham anusarichu, swabhaavasavisheshathakalaaya polynomialil prayogikkunna, kuranjathu oru eejan moolyavum egenvector undu. athinusesham 
athu yaadhaarthyamaanennu njangal kandethunnu. ippol oorthoogoonal komplimentaaya space pariganikkuka. hermitisity prakaram, oru invariyant subspace aanu. oru eejanvektar undennu kaanikkan athe aargyument prayogikkuka. parimithamaaya induction thudarnnu thelivu poortthiyaakkunnu.
spectral sidhaantham parimitha-dimensional yadhaartha aantharika uthppanna edangalile samamiti mappukalkkum baadhakamaan, ennaal oru eejanvektarinte nilanilpp beejaganithatthinte adisthaana sidhaanthathil ninnu udan pinthudarunnilla. ithu theliyikkaan, oru hermissian matricks aayi pariganikkukayum oru hermissian maatriksinte ellah eejan moolyangalum yadhaarthamaanenna vastutha upayogikkukayum cheyyuka.
aiganvektarukalude adisthaanathil matricks praathinidhyam diagonal aanu, nirmmaanatthiloode thelivu paraspara oorthoogoonal aiganvektarukalude adisthaanam nalkunnu; avaye yoonit vektarukalaayi thiranjedukkunnathiloode oraalkk aiganvektarukalude oorthoonoormal adisthaanam labikkum. spectral decomposition ennu vilikkunna jody thirichulla oorthoogoonal projakshanukalude rekheeya samyojanamaayi ezhuthaam. anuvadikkuka.
oru eejan moolyavumaayi bandhapetta eejanspace aayirikkuka. nirvachanam nirddishta eejanvektarukalude eathenkilum thiranjeduppine aasrayikkunnilla ennathu shradhikkuka. soochika eejan moolyangalekkaal vyathyasamulla edangalude oorthoogoonal nerittulla thukayaanu ithu. 
mattoru vidhathil paranjaal, oorthoogoonal projakshane soochippikkukayum athinte eejan moolyangal aanenkil, spectral vighadanam ingane ezhuthaam
a" yude spectral vighadanam formula 9 aanenkil, eathenkilum scaler formula 12-nuu formula 10 umm formula 11 umm pinthudarunnu, eathu polynomialinteyum formula 9 aanu.
shharr vighadanathinteyum eaka moolya vighadanathinteyum oru prathyeka kesaanu spectral vighadanam.
saadhaarana metriksukal.
spectral sidhaantham kooduthal pothuvaaya oru klaas metrixukalilekku vyaapikkunnu. parimithamaaya alavilulla aantharika uthppanna sthalathu oru oopparettar aakatte. undengil athu saadhaaranamaanennu parayappedunnu. ekathaanamaayi konukal thirikkaavunnathaanenkil maathrame athu saadhaaranamaanennu kaanikkan kazhiyoo. thelivu shharr vighadanathiloode, namukk eathu matrix, evide ekathaanavum mukalile thrikonaakrithiyilullathumanne ezhuthaam.
saadhaaranamaanenkil, oraal athu kaanunnu. athinaal, oru saadhaarana mukalile trikona matricks diagonal aayathinal diagonal aayirikkanam (saadhaarana matricks kaanuka). nere marichu vyakthamaan.
mattoru vidhathil paranjaal, oru unitory matricks undengil maathram athu saadhaaranamaanu.
oru diagonal matricks evideyaanu. appol, diagonalinte entrikal aigan moolyangalaanu. koolam vektarukal aiganvektarukalaanennum avaa oorthoonoormal aanennum. hermissian kesil ninnu vyathyasthamayi, entrikal yadhaarthamaayirikkanamennil.
compact selph-adjayint oopparettarmaar.
ananthamaaya alavulla hilbert speysukalude kooduthal pothuvaaya krameekaranatthil, compact selph-adjayint oopparettarmaarkkulla spectral sidhaanthathinte prasthavana parimitha-dimensional kesil ullathinu samaanamaan.
sidhaantham. oru (yadhaarthamo sankeernamo aaya) hilbert speysile oru compact selph-adjayint oopparettaraanennu karuthuka. appol aijanvektarukal ulkkollunnathin oru oorthoonoormal adisthaanam undu. oro eejan moolyavum yadhaarthamaanu.
hermissian maatriksukale sambandhichidatholam, kuranjathu oru poojyamallaatha eejanvektarinte nilanilpp theliyikkuka ennathaanu pradhana kaaryam. eejan moolyangalude nilanilpp kaanikkan oraalkk nirnnaayakangale aashrayikkaan kazhiyilla, pakshe eejan moolyangalude vyathiyana swabhaavasavisheshathakalkk samanamaaya oru maximisation aargyument upayogikkaam. 
compactness anumaanam neekkam cheythaal, oro selph-adjayint oopparettarkkum egenvectorukal undennathu "shariyalla".
parimithamaaya svayam-adjayint oopparettarmaar.
eejanvektarukalude saadhyamaaya abhavam.
nammal pariganikkunna adutha saamaanyavalkkaranam hilbert speysile parimithamaaya svayam-adjayint oopparettarmaarudethaanu. atharam oopparettarmaarkku eejan moolyangal undakanamennilla udaaharanathinu, formula _ 15 kondu gunanathinte oopparettar aayirikkam, athaayath,
eee oopparettarkku oru valia sthalathu egenvectorukal undengilum "formula _ 15" ll eejanvektarukalonnumilla. athaayath vitharana soothravaakyam _ 18, ivide formula _ 19 ennathu diarac deltta functionaanu, uchithamaaya ardhathil nirvachikkumbol ithu oru eejanvektaraanu. ennirunnaalum, dirac deltta function classical ardhathil oru functionalla, hilbert speysilo mattethenkilum baanaach speysilo allah. athinaal, deltta-functionukal formula _ 4-nte "saamaanyavalkkaricha egenvectorukal" aanu, pakshe saadhaarana ardhathil egenvectorukalalla.
spectral subspaceukalum projection moolyamulla alavukalum.
(sharee) eejanvektarukalude abhaavathil, oraalkk "eathaandu egenvectorukal" adangiya subspaceukal thirayam. udaaharanathinu, mukalil paranja udaaharanathil, formula _ 21-ll formula _ 23-nullil oru cheria edavela formula _ 22-ll pinthunaykkunna functionukalude subpace pariganichekkaam. eee space formula _ 4-nuu keezhil maattamillaathathaanu, eee sabpesile eathu formulakkum formula _ 25 formula _ 27-nuu valare aduthaanu. spectral sidhaanthathilekkulla eee sameepanathil, formula _ 4 oru parimithamaaya svayam-adjayint oopparettaraanenkil, atharam "spectral subpaseukalude" valia faamilikalkkaayi oraal thirayunnu. oro subspace anubandha projection oopparettar encode cheyyukayum ellah subspaceukaludeyum shekharanam oru projection moolyamulla alavu prathinidheekarikkukayum cheiyunnu. 
spectral sidhaanthathinte oru roopeekaranam projection moolyamulla alavumaayi bandhappett oopparettarude spectrum formula _ 29-nuu mukalilulla coordinate functionte avibhaajya gadakamaayi oopparettare prakadippikkunnu.
samshayaaspadamaaya svayam-adjayint oopparettar othungumbol, spectral sidhaanthathinte eee pathipp mukalilulla parimitha-dimensional spectral sidhaanthathinu samanamaaya onnaayi kurayunnu, oopparettar oru parimithamo ennaavunnatho aaya ananthamaaya rekheeya kompineshan oof projections aayi prakadippikkunnu ennathozhike, alavil attangal maathrame adangiyittulloo.
gunana oopparettar pathipp.
spectral sidhaanthathinte oru badal roopeekaranam parayunnath oro bounded selph-adjayint oopparettarum ekathaanamaayi oru gunana oopparettarinu thulyamaanu ennaanu. gunana oopparettarmaarkku pala tharatthil manassilakkaan eluppamaanu ennathaanu eee phalatthinte praadhaanyam.
oopparettar sidhaantham ennu vilikkunna pravarthanaparamaaya vishakalanathinte vishalamaaya gaveshana mekhalayude thudakkamaanu spectral sidhaantham; spectral alavukalum kaanuka.
hilbert speysukalil parimithamaaya saadhaarana oopparettarmaarkkaayi oru analogus spectral sidhaanthavumundu. nigamanatthile oreyoru vyathyasam ippol sankeernamaaya moolyamullathaayirikkaam ennathaanu.
nerittulla integralukal.
nerittulla samaakalanangalude adisthaanathil spectral sidhaanthathinte oru roopeekaranavumundu. ithu gunana-oopparettar formulation samaanamaan, pakshe kooduthal canonical aanu.
formula _ 4 oru parimithamaaya svayam-adjayint opparettaraayirikkatte, formula _ 32 formula _ 4-nte spectramyricute. spectral sidhaanthathinte direct-integral formulation formula _ 4-lake rund alavukale bandhappeduthunnu. aadyam, formula _ 32-ll oru alavu formula _ 35, randaamathethu, hilbert space formula _ 37-nte oru kudumbam, thudarnnu nammal direct integral hilbert space roopeekarikkunnu.
soothravaakyam _ 38
eee sthalathinte gadakangal functionukal (allengil "vibhagangal") formula 39 aanu, athaayath ellah formulakalkkum formula 40. 
spectral sidhaanthathinte nerittulla-samyojitha pathipp inipparayunna reethiyil prakadippikkaam
space formula _ 42 formula _ 4-nte "aijanspesukal" poleyaanennu karuthaam. ennirunnaalum, oru-ghadakam settu formula _ 44 nuu positteev alavu illengil, space formula _ 42 yadhaarthathil direct integralinte oru upavibhaagamalla. athinaal, formula _ 42-kall "generalised aijanspace" aayi kanakkaakkanam-athaayath, formula _ 42-nte gadakangal yadhaarthathil hilbert spacil ulppedaatha "aijanvektarukal" aanu.
spectral sidhaanthathinte gunana-oopparettarum nerittulla avibhaajya soothravaakyangalum oru svayam-adjayint oopparettare gunana oopparettarinu ekathaanamaayi thulyamaayi prakadippikkunnundel, nerittulla avibhaajya sameepanam kooduthal canonical aanu. onnamathaayi, direct integral nadakkunna settu (oopparettarude spectrum) canonical aanu. randaamathaayi, nammal gunikkunna function direct-integral sameepanathil canonical aannu function formula _ 48.
chaakrika vektarukalum lalithamaaya spectravum.
vector formula _ 51 hilbert spacinte saandramaaya upavibhagam vyaapichukidakkukayaanel vector formula _ 25 neey formula _ 4-nte chaakrika vector ennu vilikkunnu. formula _ 4 oru cyclic vekttar nilanilkkunna oru bounded selph-adjayint oopparettaraanennu karuthuka. anganeyaanengil, spectral sidhaanthathinte direct-integral, gunanam-oopparettar formulationukal thammil vyathyasamilla. vaasthavathil, aaa saahacharyathil, formula _ 4-nte spectrum formula _ 29-ll oru alavu formula _ 35 undu, athaayath formula _ 4 ekathaanamaayi formula _ 58-leeii "formula _ 41 oopparettar kondu gunikkunnathin" thulyamaanu. eee phalam formula _ 4-neey oresamayam oru gunana oopparettaraayum "" direct integral "aayum prathinidheekarikkunnu, kaaranam formula _ 58 oru nerittulla avibhaajyaghadakam maathramaan, athil oro hilbert space formula _ 42 umm verum formula _ 62 aanu.
oro bounded selph-adjayint oopparettarum oru cyclic vector amgeekarikkunnilla; theerchayaayum, direct integral decompositionile prathyekathayaal, ellah formulakalkkum _ 42-kalkkum alavu onnu ullappol maathrame ithu sambhavikkoo. ithu sambhavikkumbol, spectral multiplisity sidhaanthathinte ardhathil formula _ 4 nuu "lalithamaaya spectrum" undennu nammal parayunnu. athaayath, oru chaakrika vektarine amgeekarikkunna oru parimithamaaya svayam-adjayint oopparettare vyathyasthamaaya eejan moolyangalulla (i.iii.oro eejan moolyathinum onniladhikam gunithangalundu).
oro formula oru chaakrika vector amgeekarikkunnillenkum, 4 enna formulakku chaakrika vector ulla asthiramaaya subspaceukalude nerittulla thukayayi hilbert space vighadippikkaan kazhiyumennu kaanaan eluppamaanu. eee nireekshanam spectral sidhaanthathinte gunana-pravartthakanteyum nerittulla-avibhaajya roopangaludeyum thelivukalude thacolaanu.
functional calculus.
spectral sidhaanthathinte (eathu roopathilum) oru pradhana prayogam oru functional calculus nirvachikkuka ennathaanu. athaayath, formula _ 4-nte spectrathil nirvachichirikkunna oru function formula _ 67 nalkiyaal, oru oopparettar formula _ 69 nirvachikkaan nammal aagrahikkunnu. formula _ 67 oru positteev power aanenkil, formula _ 71, formula _ 69 formula _ 4, formula _ 75-nte formula _ 73 power maathramaan. rasakaramaaya kesukal formula _ 67 oru square root allengil exponential polulla nonpolinomiyal functionaanu. spectral sidhaanthathinte eathenkilum pathippukal atharamoru functional calculus nalkunnu. direct-integral pathippil, udaaharanathinu, formula _ 69 direct integralile "formula _ 67 kondu gunikkuka" oopparettaraayi pravartthikkunnu
athaayath, direct integralile oro space formula _ 42 ennathu _ 69 enna formulayude (saamaanyavalkkaricha) eejanspace aanu, eejan value formula _ 82 aanu.
general selph-adjoyint oopparettarmaar.
deferential oopparettarmaar polulla vishakalanathil sambhavikkunna niravadhi pradhana leanier oopparettarmaar paridhiyillaathavaraanu. eee sandarbhangalil svayam-adjayint oopparettarmaarkku baadhakamaaya oru spectral sidhaanthavumundu. oru udaaharanam nalkukayaanengil, oro sthira-gunakam deferential oopparettarum ekathaanamaayi oru gunana oopparettarinu thulyamaanu. vaasthavathil, eee thulyatha nadappilaakkunna unitory oopparettar forrier parivarthanamaanu; gunana oopparettar oru tharam forrier gunakamaanu.
pothuve, svayam-adjayint oopparettarmaarkkulla spectral sidhaantham niravadhi thulya roopangal eduthekkaam. prathyekichum, muun vibhaagathil parimithamaaya svayam-adjayint oopparettarmaarkkaayi nalkiyirikkunna ellah formulationukalum-projection-moolyamulla alavukolukal, gunana-oopparettar pathipp, direct-integral pathipp-paridhiyillaattha svayam-adjayint oopparettarmaarkkaayi thudarunnu, domaine prasnangal kaikaaryam cheiyunnathinulla cheria saankethika parishkkaranangalode.