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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Huupponen, Tuukka", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-05-25T06:15:16Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-05-25T06:15:16Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/87184", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa perehdyt\u00e4\u00e4n matriisiteoriaan. Tarkastelu keskittyy neli\u00f6matriiseihin, niiden ominaisarvoihin ja niit\u00e4 vastaaviin ominaisvektoreihin. Tarkastelu rajataan kahteen osaan, joista toiseen esitet\u00e4\u00e4n k\u00e4yt\u00e4nn\u00f6nsovellus, jossa tarkastellaan Googlen hakukoneen toiminnan matemaattista taustaa. \n\nVaadittavien lineaarialgebran esitietojen j\u00e4lkeen tarkastellaan positiivisia neli\u00f6matriiseja. Ensin esitell\u00e4\u00e4n aputuloksia liittyen positiivisiin matriiseihin, sek\u00e4 niiden itseisarvoltaan suurimpaan ominaisarvoon eli spektraalis\u00e4teeseen ja t\u00e4t\u00e4 ominaisarvoa vastaavaan ominaisvektoriin. N\u00e4iden aputulosten tarkastelun j\u00e4lkeen esitell\u00e4\u00e4n Perronin lause ja sille esitet\u00e4\u00e4n todistus. Perronin lause toteaa, ett\u00e4 positiivisella neli\u00f6matriisilla on itseisarvoltaan suurin ominaisarvo, joka on aidosti suurempi kuin mik\u00e4\u00e4n muu matriisin ominaisarvojen itseisarvoista. Lis\u00e4ksi lause toteaa, ett\u00e4 t\u00e4m\u00e4n itseisarvoltaan suurimman ominaisarvon algebrallinen kertaluku on 1 ja sit\u00e4 vastaava ominaisvektori on positiivinen.\n\nPerronin lauseen todistuksen j\u00e4lkeen perehdyt\u00e4\u00e4n verkkoteoriaan, jonka avulla havainnollistetaan graafisesti tutkielman sovellusta. Verkkoteoriaan liittyvien m\u00e4\u00e4ritelmien ja tulosten esitt\u00e4misen j\u00e4lkeen tarkastellaan differenssiyht\u00e4l\u00f6it\u00e4. Differenssiyht\u00e4l\u00f6iss\u00e4 ajan hetki $t$ liitet\u00e4\u00e4n ajanhetkeen $t+1$ lineaarisesti siten, ett\u00e4 $Av_t=v_{t+1}$, miss\u00e4 neli\u00f6matriisia $A$ kutsutaan siirtym\u00e4matriisiksi.\n\nDifferenssiyht\u00e4l\u00f6iss\u00e4 keskityt\u00e4\u00e4n Markovin ketjuihin, jotka ovat todenn\u00e4k\u00f6isyyksi\u00e4 kuvaavia differenssiyht\u00e4l\u00f6it\u00e4. T\u00e4llaisten differenssiyht\u00e4l\u00f6iden siirtym\u00e4matriisit ovat stokastisia matriiseja, jotka ovat ei-negatiivisia matriiseja ja niiden jokainen sarake summautuu luvuksi yksi. Stokastisten matriisien lis\u00e4ksi Markovin ketjuissa esiintyv\u00e4t vektorit $v_i$ ovat todenn\u00e4k\u00f6isyysvektoreita eli ei-negatiivisia vektoreita, joiden alkiot summautuvat luvuksi yksi.\n\nMarkovin ketjujen teoria toimii teoriapohjana Googlen Pagerank-algorit-mille, jossa sovelletaan Perronin lausetta. M\u00e4\u00e4ritell\u00e4\u00e4n Google-matriisi, joka on positiivinen stokastinen matriisi. Perronin lause takaa, ett\u00e4 t\u00e4m\u00e4n matriisin spektraalis\u00e4de on luku 1 ja sen viritt\u00e4\u00e4 positiivinen ominaisvektori, jota kutsutaan tasapainotilavektoriksi. Pagerank-algoritmissa tarkastellaan Markovin ketjua, jonka m\u00e4\u00e4r\u00e4\u00e4 Google-matriisi $G$ ja tasapainotilavektori $w$. T\u00e4ll\u00f6in Markovin ketjun $Gv_t=v_{t+1}$ avulla saadaan internetin nettisivujen t\u00e4rkeysj\u00e4rjestys eli j\u00e4rjestys, jossa hakutulokset tulevat n\u00e4kyviin hakukoneessa. T\u00e4m\u00e4n t\u00e4rkeysj\u00e4rjestyksen kertoo Markovin ketjun vektori $v_{t+1}$.\n\nTutkielman lopuksi tarkastellaan Perronin lauseen yleistyst\u00e4 eli Perronin ja Frobeniuksen lausetta. T\u00e4m\u00e4 lause antaa vastaavat tulokset kuin Perronin lause, koskien ei-negatiivisia redusoitumattomia matriiseja. \n\nTarkoituksena on esitell\u00e4 Perronin ja Frobeniuksen lauseen yleinen versio. T\u00e4m\u00e4n j\u00e4lkeen esitell\u00e4\u00e4n ja todistetaan kyseisen lauseen erikoistapaus, jossa tarkastellaan ei-negatiivisia redusoitumattomia neli\u00f6matriiseja, joiden jokin positiivinen kokonaislukupotenssi on positiivinen neli\u00f6matriisi. T\u00e4m\u00e4 erikoistapaus toteaa vastaavat tulokset, kuin yleinen Perronin ja Frobeniuksen lause.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2023-05-25T06:15:16Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-05-25T06:15:16Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Perronin ja Frobeniuksen lause", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202305253248", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differenssiyht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lineaarialgebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "normit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkkoteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ominaisarvot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Markovin ketjut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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