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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Backlund, Mika", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-07T07:23:07Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-07T07:23:07Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95634", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on k\u00e4sitell\u00e4 pseudok\u00e4\u00e4nteismatriiseja ja pienimm\u00e4n neli\u00f6summan menetelm\u00e4\u00e4 sek\u00e4 niiden taustalla olevaa lineaarialgebran teoriaa. Pienimm\u00e4n neli\u00f6summan menetelm\u00e4 on tutkielman huipentuma ja sit\u00e4 ennen perehdyt\u00e4\u00e4n laajasti lineaarialgebran teorioihin. Tutkielma alkaa lineaarialgebran perusm\u00e4\u00e4ritelmill\u00e4 luoden samalla pohjaa my\u00f6hemmin k\u00e4sitelt\u00e4ville aiheille. \n\nTutkielman ensimm\u00e4inen merkitt\u00e4v\u00e4 teema on matriisin singulaariarvohajotelma, jonka mukaan jokainen \u00e4\u00e4rellisulotteinen matriisi on esitett\u00e4viss\u00e4 kolmen matriisin tulona. Tullaan todistamaan v\u00e4ite, ett\u00e4 singulaariarvohajotelma on olemassa. Nimi singulaariarvohajotelma on tullut siit\u00e4, ett\u00e4 yksi hajotelman matriiseista sis\u00e4lt\u00e4\u00e4 alkuper\u00e4isen matriisin singulaariarvot. \n\nToinen merkitt\u00e4v\u00e4 teema on pseudok\u00e4\u00e4nteismatriisi, jonka olemassa olo ja yksik\u00e4sitteisyys todistetaan. Tullaan huomaamaan, ett\u00e4 pseudok\u00e4\u00e4nteismatriisin ja singulaariarvohajotelman v\u00e4lill\u00e4 on kytk\u00f6s; singulaariarvohajotelman matriiseista saadaan muodostettua pseudok\u00e4\u00e4nteismatriisi. Lis\u00e4ksi k\u00e4sitell\u00e4\u00e4n pseudok\u00e4\u00e4nteismatriisin perusominaisuuksia sek\u00e4 lohkotun matriisin pseudok\u00e4\u00e4nteismatriisia. \n\nTutkielman lopussa p\u00e4\u00e4st\u00e4\u00e4n k\u00e4sittelem\u00e4\u00e4n pienimm\u00e4n neli\u00f6summan menetelm\u00e4\u00e4. M\u00e4\u00e4ritell\u00e4\u00e4n pienimm\u00e4n neli\u00f6summan ratkaisu ja otetaan k\u00e4ytt\u00f6\u00f6n kaksi laskutapaa, jolla pienimm\u00e4n neli\u00f6summan ratkaisu saadaan. Ensimm\u00e4inen laskutapa soveltuu niin sanotuille normaaleille yht\u00e4l\u00f6ryhmille ja j\u00e4lkimm\u00e4inen laskutapa soveltuu yleisesti kaikille yht\u00e4l\u00f6ryhmille. J\u00e4lkimm\u00e4inen laskutapa kytkeytyy pseudok\u00e4\u00e4nteismatriisiin siten, ett\u00e4 sen laskukaavassa esiintyy kyseinen matriisi. Pienimm\u00e4n neli\u00f6summan menetelm\u00e4\u00e4 soveltaen saadaan erilaisille pisteaineistoille m\u00e4\u00e4ritetty\u00e4 paras mahdollinen kuvaaja.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-06-07T07:23:07Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-07T07:23:07Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "46", "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": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "pienimm\u00e4n neli\u00f6summan menetelm\u00e4", "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": "pseudok\u00e4\u00e4nteismatriisi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lohkomatriisi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Pseudok\u00e4\u00e4nteismatriisit ja pienimm\u00e4n neli\u00f6summan menetelm\u00e4", "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-202406074396", "language": null, "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": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "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.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": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kompleksiluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lineaarialgebra", "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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