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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": "Kirsil\u00e4, Jaakko", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-02-11T06:24:12Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-02-11T06:24:12Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/74092", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on esitell\u00e4 ja todistaa matriisin singulaariarvohajotelma, jonka mukaan jokainen m x n matriisi A voidaan esitt\u00e4\u00e4 muodosssa A=USV^T, miss\u00e4 matriisit U ja V ovat ortogonaalisia ja S on diagonaalimatriisi. Tuloksen muotoa voidaan verrata matriisin diagonalisoituvuuteen. Diagonalisoituvuudesta puhuttaessa matriisin t\u00e4ytyy kuitenkin olla neli\u00f6matriisi ja lis\u00e4ksi kaikki neli\u00f6matriisit eiv\u00e4t ole diagonalisoituvia. Singulaariarvohajotelma on olemassa kaikille m x n matriiseille. Tutkielmassa tarvittavia tuloksia esitell\u00e4\u00e4n tutkielman alussa lineaarialgebran ja matriisiteorian kannalta merkitt\u00e4vien nelj\u00e4n aliavaruuden avulla. Lis\u00e4ksi tutustutaan n\u00e4iden aliavaruuksien rooliin matriisin A toiminnassa. Singulaariarvohajotelmaan liittyy olennaisena osana matriisin A singulaariarvot, jotka ovat matriisin A^TA ominaisarvojen neli\u00f6juuret. Ennen singulaariarvohajotelman esittely\u00e4 tutkielmassa tutustutaankin matriisin A^TA ominaisuuksiin, joita tarvitaan my\u00f6hemmin singulaariarvohajotelman todistuksessa. Tarkoituksena on my\u00f6s esitell\u00e4 matriisin singulaariarvohajotelman sovelluksia. Ensimm\u00e4isen\u00e4 sovelluksena on matriisin pseudoinverssi, jota voidaan pit\u00e4\u00e4 k\u00e4\u00e4nteismatriisin yleistyksen\u00e4. Pseudoinverssin avulla voidaan ratkaista lineaarisia yht\u00e4l\u00f6ryhmi\u00e4, jotka ovat muotoa Ax=b. T\u00e4ll\u00e4 yht\u00e4l\u00f6ryhm\u00e4ll\u00e4 ei kuitenkaan aina ole ratkaisua. T\u00e4ll\u00f6in voidaan kuitenkin etsi\u00e4 vektoria x\u2019, joka minimoi lausekkeen ||b-Ax\u2019||^2. T\u00e4t\u00e4 vektoria x\u2019 kutsutaan pienimm\u00e4n neli\u00f6summan ratkaisuksi. Osoittautuu, ett\u00e4 pseudoinverssin avulla l\u00f6ydet\u00e4\u00e4n my\u00f6s pienimm\u00e4n neli\u00f6summan ratkaisu. Singulaariarvohajotelman sovelluksena on my\u00f6s matriisin approksimointi alemman asteen matriisilla. Tarkoituksena on esitell\u00e4 tulos, jonka mukaan matriisin singulaariarvohajotelman avulla matriisille saadaan paras alemman asteen approksimaatio Frobenius normin suhteen. Tutkielman lopuksi sovelletaan matriisin alemman asteen approksimaatiota valokuvan h\u00e4vi\u00f6llisess\u00e4 pakkaamisessa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-02-11T06:24:12Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-02-11T06:24:12Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "58", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "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": "matriisin singulaariarvohajotelma", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "singulaariarvot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "pseudoinverssi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "pienimm\u00e4n neli\u00f6summan ratkaisu", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisin approksimointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kuvan h\u00e4vi\u00f6llinen pakkaaminen", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matriisin singulaariarvohajotelma", "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-202102111528", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "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.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": 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