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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Koskim\u00e4ki, Saaga", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-10-06T08:44:19Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-10-06T08:44:19Z", "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/123456789/89477", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tarkastella matriisin kolmea erilaista hajotelmaa. Matriisihajotelmien avulla matriisi voidaan esitt\u00e4\u00e4 hy\u00f6dyllisess\u00e4 muodossa muita tuloksia varten. Tutkielmassa perehdyt\u00e4\u00e4n matriisin CR-hajotelmaan, symmetrisen matriisin diagonaalihajotelmaan sek\u00e4 singulaariarvohajotelmaan. Lis\u00e4ksi singulaariarvohajotelman sovelluksena k\u00e4sitell\u00e4\u00e4n matriisin perturbaatiota.\nMatriisin CR-hajotelmassa matriisi A esitet\u00e4\u00e4n matriisien C ja R avulla muodossa A = CR. T\u00e4ss\u00e4 hy\u00f6dynnet\u00e4\u00e4n matriisin A astetta, lineaarisesti riippumattomia\nsarakkeita sek\u00e4 redusoitua porrasmatriisia. Tutkielmassa tarkastellaan my\u00f6s, miten\nsymmetrinen matriisi A voidaan ortogonaalisesti diagonalisoida diagonaalimatriisin\nD ja ortogonaalisen matriisin V avulla muodossa A=VDV^T. T\u00e4t\u00e4 kutsutaan symmetrisen matriisin diagonaalihajotelmaksi. Hajotelmassa matriisin D diagonaalialkiot\nkoostuvat matriisin A ominaisarvoista ja matriisi V kyseisi\u00e4 ominaisarvoja vastaavista ominaisvektoreista.\nKolmantena hajotelmana tutkielmassa k\u00e4sitell\u00e4\u00e4n matriisin A singulaariarvohajotelmaa A = U\u03a3V^T, miss\u00e4 matriisi \u03a3 on singulaariarvoista koostuva diagonaalimatriisi ja matriisit U ja V ovat ortogonaalisia. Lis\u00e4ksi singulaariarvohajotelman avulla\ntodistetaan matriisin perturbaatiolause, jossa tarkastellaan matriisin A+cd^T astetta,\nkun t\u00e4ss\u00e4 matriisin cd^T aste on yksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-10-06T08:44:19Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-10-06T08:44:19Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "55", "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.subject.other", "value": "matriisihajotelmat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "singulaariarvohajotelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisin CR-hajotelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "symmetrinen matriisi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisin perturbaatio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matriisihajotelmia", "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-202310065507", "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": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in 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": "matriisit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "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": "matriisilaskenta", "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": "vektorit (matematiikka)", "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": 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