bahupadi matricks varnakrameey ghatak

bahupadi matrixcha pranali siddhant aanee niyantran siddhant yaa kshetrat mothya pramanavar abhyas keela jaato aanee sthir bahupadanshi sambandhit itar upayog pahile aahet. sthirata siddhantat, dwibhinnata sthir bahupadi aanee vastavik shoonya bahupadisathi nirdharak matricks pratinidhitva shodhanyasathi varnakrameey ghatak vaparlya gele aahet. yacha abhyas karnyasathi vaparale jaanare eka pramukh sadhan mhanje bahupadi matricks varnakrameey ghatak kinwa matricks phasor-riz pramey mhanoon olakhale jaanare matricks ghatak.
konatyahi vastavik input sutrasathi nakaratmak nasaleli moolye ghenari bahupadi, eka eka avibhajya sakaratmak bahupadi sootra _ 1 dile tarr, phasor-riz pramey bahupadi varnakrameey gunank sootra _ 3 dete. yaa swarupachya parinamanna samanyatah sakaratmak stellensatz mhanoon sambodhale jaate. sakaratmak nishchitata hii sakaratmakateche matricks analogue manoon, bahupadi matricks spectral factorization heey sakaratmak nishchit shreni asalelya bahupadi matrixsathi samaan factorization pradan karate. heey vighatan scalar matricks sootra _ 4 sathichya cholesky vighatanshi dekhil sambandhit aahe. haah parinam moolatah vayanardvare adhik samanya sandarbhat siddha jhaala hota joou integrable matricks-moolya asalelya karyanshi sambandhit hota jyamadhye integrable logg determinant dekhil hota. kaaran anuprayog bahutekda bahupadi nirbandhanshi sambandhit asatat, yaa prakarnavar laksh kendrit karnare sope purave aanee vaiyaktik vishleshan astitvat aahe. visheshtah bahupadi matrixchi vastavik ardh-bijganitiy upasanchanvar sakaratmak nishchit pratima asate tevha vicharat gheuun, kamkuvat sakaratmak-stellensatz paristhiticha abhyas keela gela aahe. alikadech anek prakashananni yaa sambandhit parinamansathi puravyanna suvyavasthit karnyavar laksh kendrit kele aahe. haah lekh sadharanpane lasha ephremidzchya alikadil purava paddhatiche anusaran karato jee keval prathamik rekheey bijganit aanee jatil vishleshanavar avalambun aahe.
resheey-chaturthansh-gossian niyantranat varnakrameey ghatak mothya pramanat vaparale jaatat. yaa anuprayogamule varnakrameey ghatakanchi ganana karnyasathi anek algoridam tayar jhaale aahet. kahee aadhunik algoridam moolatah winerney abhyasalelya adhik samanya mandanivar laksh kendrit karatat. sootra _ 5 madhye yaa samasyela bahupadi varnakrameey ghatak kinwa phasor-riz pramey mhanoon olakhale jaate aanee tyat anek shastriy algoridam aahet. kahee aadhunik algoridammadhe ghatakanchya gananela gatee denyasathi toplitz matricks pragatincha vapar keela aahe.
vidhan.
sootra _ 6 heey bahupadi matricks asoo dya jithe pratyek pravesh sootra _ 7 heey bahutek sootra _ 8 varr degreeche eka jatil gunank bahupadi aahe. samaja jawaljawal serve sootra _ 9 sathee sootra _ 10 heey eka sakaratmak nishchit hermician matricks aahe. mug eka bahupadi matricks sootra _ 11 astitvat aahe, jase qee serve sootra _ 9 sathee sootra _ 12. pudhe aapan sootra _ 11 shodhu shakato, jeey khalchya ardhya pratalavar ekras nahee.
guntaguntichya inputparyant vistar.
lakshat ghya qee jarr sootra 15 asel tarr sootra 16. jevha sootra 17 heey eka jatil gunank bahupadi kinwa jatil gunank tarksangat karya asate tevha aaplyakade sootra 18 dekhil anukrame bahupadi kinwa tarksangat karya aahe. sootra _ 9 sathee aaplyakade heey formula _ 20 aahe kaaran khaleel nirikshan aaheh sootra _ 21 aanee sootra _ 10 chya nondi guntaguntichya bahupadi asalyane jyaa vastavik reshevar sahmat aahet, tyaa pratyakshat samaan bahupadi aahet. aapan assa nishkarsh kadhu shakato qee teey serve guntaguntichya mahitisathi sahmat aahet.
tarksangat varnakrameey ghatak.
sootra _ 1 heey eka tarksangat bahupadi karya asoo dya jithe jawaljawal serve sootra _ 9 sathee sootra _ 24. nantar sootra _ 3 saha tarksangat sootra _ 26 astitvat aahe jethe sootra _ 26 laa khalchya ardhya pratalat konatehi dhruv kinwa shoonya naheet. heey vighatan norm formula _ 29 chya complexes skelardware gunakar karnyaparyant adviteey aahe. heey formula _ 5 prakaranaparyant maryadit bahupadi matricks spectral factorization prameyachya vidhanashi sambandhit aahe.
astitva siddha karnyasathi sootra 31 liha jethe sootra 32. sootra 33 dett asatana, aapan assa nishkarsh kadhu shakato qee sootra 34 vastavik aanee sakaratmak aahe. sootra _ 35 neey vibhajit karoon aapan monic kesamadhye qamee karato. ansh aanee vibhajkamadhye mulanche vegvegale sanch asatat, mhanoon donhimadhye donhimadhye serve vastavik mulanmadhye summ gunakar asane aavashyak aahe (sthanik patalivar chinh badal talhanyasathi). sootra _ 1 madhye fakt guntaguntichi mule aanee khamb aahet asha sthiteet qamee karnyasathi aapan yaa vastavik mulanche vibhajan karoo shakato. gruhitkanusar aaplyakade sootra 37 aahe. kaaran serve sootra 38 heey jatil aahet (aanee mhanunach sanyugmanache nishchit bindu naheet) teey donhi sanyugma jodyanmadhye yetat. pratyek sanyugmit jodisathi, varchya ardhya pratalatil shoonya kinwa khamb nivada aanee sootra _ 26 prapt karnyasathi teey jamaa qaraa. advitiyata parinam pramanit paddhatine anusaran karato.
cholesky vighatan.
yaa parinamachi prerana eka gunank aahe jeey sakaratmak nishchit matrixche vaishishtya darshavate.
scalar matrixsathi vighatan.
konatehi sakaratmak nishchit scalar matricks sootra _ 40 dile tarr, cholesky vighatan aaplyala sootra _ 41 lihinyachi parvanagi dete jethe sootra _ 42 haah eka nimn trikoni matricks aahe. jarr aapan khalchya trikoni matrixparyant maryadit nasel tarr aapan formula _ 43 chya serve factorization vichar karoo shakato. formula _ 45 dware yogya gunakarakhali formula _ 42 chya kakshet pahoon serve factorization sadhya kele jaatat qee nahee heey tapasane katheen nahee.
khalchya trikoni apghatan prapt karnyasathi aapan pahilya panktiche aanee pahilya stambhache vibhajan karoon samavisht karatoh sootra _ 46 heey sootra _ 47 chya sandarbhat sodavalyas aaplyala formula _ 48 formula _ 49 formula _ 50 milate.
sootra 40 heey nishchitpane sakaratmak asalyane, aaplyakade sootra 52 hii eka sakaratmak vastavik sankhya aahe, mhanoon tyache vargamool aahe. ujavya hatachya bajoone inductionchi shevatchi aut mhanje sootra _ 40 chey shoor purak aahe, jeey swatach sakaratmak nishchit aahe.
tarksangat matrixsathi vighatan.
aata sootra _ 6 chaa vichar qaraa jithe sootra _ 7 hii jatil tarksangat karye aahet aanee sootra _ 10 heey jawaljawal serve vastavik sootra _ 2 sathee sakaratmak nishchit hermician aahe. nantar aapan varr kelelya samamiteey gossian nirmulanadware, aaplyala fakt eka tarksangat sootra _ 58 astitvat aahe heey darshavine aavashyak aahe, jeey vastavik sootra _ 2 sathee sootra _ 59 aahe, jeey aaplya tarksangat varnakrameey ghatakatoon yete. ekada teey midale qee aapan sootra _ 61 sathee nirakaran karoo shakato. kaaran dhruvanpasun door asalelya vastavik sootra _ 2 sathee shoor purak sakaratmak nishchit aahe aanee shoor purak haah eka tarksangat bahupadi matricks aahe joou aapan sootra _ 63 shodhanyasathi samavisht karoo shakato.
aaplyala pratyakshat sootra 64 milate qee nahee heey tapasane katheen nahee, jethe sootra 65 heey eka tarksangat bahupadi matricks aahe jyachya khalchya ardhya pratalat konatehi dhruv naheet.
bahupadi vighatananparyant vistar.
bahupadi matricks varnakrameey ghatakanache astitva siddha karnyasathi, aapan tarksangat bahupadi matricks cholesky vighatanapasun suruvat karato aanee khalchya ardhya pratalachi ekatmata kadhoon taknyasathi tyat badal karato. mhanjech, pratyek pravesh sutrasah sootra _ 6 dile gele aahe, eka jatil gunank bahupadi, vastavik sootra _ 2 sathee sootra _ 64 saha tarksangat bahupadi matricks sootra _ 65 aahe, jethe sootra _ 65 laa nimn ardha samtal dhruv naheet. vastavik sootra _ 2 sathee ekatmak moolya asalele tarksangat bahupadi matricks sootra _ 72 dile tarr, aanakhi eka vighatan astitvat aahe, sootra _ 74.
khalchya ardhya-samtal ekatmata kadhoon takane.
jarr sootra _ 75 asel tarr eka scalar ekatmak matricks sootra _ 76 astitvat aahe, jase qee sootra _ 77. heey sootra soochit karate qee _ 78 haah pahila stambh sootra _ 79 varr nahisa jhaala aahe. sootra _ 79 vareel ekatmata kadhoon taknyasathi aapan konatyahi nondinchya khalchya ardhya pratalat konatehi dhruv naan aanata, a varr eka qamee shoonyasah (gunakarane) sootra _ 81 formula _ 82 haah nirdharak gunakar karato.
serve see paryant vishleshan vadhva.
badalannantar, vighatan sootra _ _ sootra _ 11 heey vishleshanatmak aanee khalchya ardhya pratalavar ulat karata yenyajoge aahe. varchya ardhya pratalaparyant vishleshanatmakata vadhavanyasathi aaplyala heey mukhya nirikshan aavashyak aaheh eka tarksangat matricks sootra dile aahe jeey khalchya ardhya pratalamadhye vishleshanatmak aahe aanee khalchya ardhya pratalamadhye ekras nahee, aamchyakade sootra aahe _ 86 heey khalchya ardhya pratalamadhye vishleshanatmak aanee ekras nahee. vishleshanatmakata hii sanlagn matricks sutravarun yete (kaaran sootra _ 11 aanee sootra _ 88 yaa donhi nondi khalchya ardhya pratalavar vishleshanatmak aahet). sootra 89 pasoon nonsingularity yete, jyamadhye sootra 90 madhye dhruv asalelya thikani fakt shoonya asoo shakatat. tarksangat bahupadi matrixchya nirdharkamadhye keval dhruv asoo shakatat jethe tyachya nondinna dhruv asatat, mhanoon sootra _ 90 laa khalchya ardhya pratalat dhruv nasatat.
aaplya nirikshanapasun teey guntaguntichya inputparyant, serve guntaguntichya sankhyansathi sootra _ 92 aahe. heey sootra 93 soochit karate. sootra 86 heey khalchya ardhya pratalavar vishleshanatmak asalyane, sootra 11 heey varchya ardhya pratalavar vishleshanatmak aahe. shevati jarr sootra _ 11 chaa vastavik reshevar dhruv asel tarr sootra _ 97 chaa vastavik reshevar samaan dhruv aahe joou sootra _ 10 chya vastavik reshevar konatehi dhruv naheet (gruhitkanusar teey sarvatra vishleshanatmak aahe).
varilvaroon asey disun yete qee jarr sootra _ 11 heey vishleshanatmak aanee khalchya ardhya pratalavar ulat karata yenyajoge asel tarr sootra _ 11 heey sarvatra vishleshanatmak aahe aanee mhanoon eka bahupadi matricks aahe.
advitiyata.
done bahupadi matricks vighatan dile gele aahet jeey khalchya ardhya planformula _ 101 varr ulat karata yetat. aapan sootra _ 102 madhye punarvyavasthit karato. sootra _ 103 khalchya ardhya pratalavar vishleshanatmak aanee ekras nasalele asalyane, sootra _ 104 haah eka tarksangat bahupadi matricks aahe joou khalchya ardhya pratalavar vishleshanatmak aanee ulat karata yeto. mug vareel tyach yuktivadanusar, aaplyakade sootra 104 heey pratyakshat eka bahupadi matricks aahe jeey serve vastavik sutrasathi ekatmak aahe. yacha arth assa qee jarr sootra 107 heey sootra _ 11 chey sootra _ 108 weey roow asel tarr sootra _ 110. vastavik sootra _ 2 sathee hii naun-negative bahupadichi berries aahe jee sthirankat berries karate, yacha arth pratyek berries pratyakshat sthir bahupadi aahe. mug sootra _ 112 jethe sootra _ 76 heey eka scalar ekatmak matricks aahe.
udaharanadakhal.
sootra 114 chaa vichar qaraa. mug samamiteey gossian nirmulanadware aaplyala tarksangat vighatan sootra 115 milate. yaa vighatanala varchya ardhya pratalat konatehi dhruv naheet. tathapi, nirdharak sootra 116 aahe, mhanoon aaplyala sootra 117 vareel ekatmata door karnyasathi aaplya vighatanamadhye badal karane aavashyak aahe. pratham aapan scalar ekatmak matrixney gunakar karoon sootra 108 varr stambh nahisa karato. sootra 119 chaa vichar qaraa. nantar aaplya vighatan formulasathi eka naveen umedvar aahe. aata pahila stambh yethe nahisa hoto
sootra _ 108, mhanoon aapan sootra _ 123 soochana sootra _ 124 prapt karnyasathi sootra _ 122 dware (ujavikade) gunakar karato. heey aapale ichchit vighatan sootra _ 125 aahe jyat khalchya ardhya pratalat konatihi ekakata nahee.

