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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kajan, Minttu", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-06-06T06:35:21Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-06-06T06:35:21Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/103138", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu-tutkielmassa perehdyt\u00e4\u00e4n lineaarialgebraan ja erityisesti ominaisarvoteoriaan hieman tavanomaisesta poikkeavasta n\u00e4k\u00f6kulmasta. Tutkielman tavoitteena on esitell\u00e4 erilainen l\u00e4hestymistapa ominaisarvoteorian perusteisiin. Lineaarialgebran kursseilta ja oppikirjoista tuttu tapa todistaa ominaisarvoihin liittyvi\u00e4 tuloksia on k\u00e4ytt\u00e4\u00e4 determinanttia. T\u00e4ss\u00e4 tutkielmassa tutustutaan vaihtoehtoiseen tapaan, jossa determinantteja ei tarvita. \n\nTutkielman alussa k\u00e4sitell\u00e4\u00e4n joitakin oleellisia esitietoja ja t\u00e4m\u00e4n j\u00e4lkeen l\u00e4hdet\u00e4\u00e4n rakentamaan teoriaa, jonka avulla voidaan todistaa ominaisarvoihin, ominaisvektoreihin ja yleistettyihin ominaisvektoreihin liittyvi\u00e4 tuloksia. Keskeisen\u00e4 osana rakennettavaa teoriaa on kompleksisten polynomien soveltaminen lineaarikuvauksiin. \n\nTutkielmassa k\u00e4sitell\u00e4\u00e4n my\u00f6s vektoriavaruuksien suoria summia, jotka ovat keskeisess\u00e4 osassa viidennen ja kuudennen eli viimeisen luvun teoriassa. Viidennen luvun p\u00e4\u00e4tuloksena todistetaan lause, jota k\u00e4ytt\u00e4en voidaan tutkia lineaarikuvausten rakennetta yleistettyjen ominaisavaruuksien ja suorien summien avulla. Viimeisess\u00e4 luvussa sovelletaan viidennen luvun p\u00e4\u00e4tulosta ja osoitetaan sen avulla, ett\u00e4 jokaiselle lineaarikuvaukselle n-ulotteisesta \u00e4\u00e4rellisest\u00e4 kompleksisesta vektoriavaruudesta itselleen l\u00f6ytyy kanta, jossa sit\u00e4 vastaava matriisi on lohkodiagonaalinen matriisi, jonka lohkot ovat yl\u00e4kolmiomatriiseja. Jokainen lohko vastaa yht\u00e4 lineaarikuvauksen ominaisarvoa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-06-06T06:35:21Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-06-06T06:35:21Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "40", "language": null, "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": "ominaisvektorit", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Ominaisarvoteoriaa ilman determinantteja", "language": null, "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-202506064944", "language": null, "element": "identifier", "qualifier": "urn", "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": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "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": "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": "fi", "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.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": "ominaisarvot", "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"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}]
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