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[{"key": "dc.contributor.author", "value": "Lindberg, Antti", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2014-08-25T13:22:51Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2014-08-25T13:22:51Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2014", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1444589", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/44091", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman sis\u00e4lt\u00f6 voidaan karkeasti jakaa kahteen osaan. Ensimm\u00e4isess\u00e4 on tarkoituksena tarkastella polynomimatriiseja ja erityisesti osoittaa toimiviksi kaksi niiden muokkaamiseen soveltuvaa algoritmia. Algoritmit toimivat osittain samalla idealla kuin lineaarialgebran perusteista tuttu Gaussin ja Jordanin menetelm\u00e4. Polynomit tuovat menetelmiin kuitenkin uutta sis\u00e4lt\u00f6\u00e4 erityisesti jaollisuusominaisuuksiensa vuoksi. Tarkasteltavat matriisit ovat aina neli\u00f6matriiseja, ja polynomien kerroinkunnan karakteristika oletetaan nollaksi. \n\nEnsimm\u00e4inen algoritmi osoittaa, ett\u00e4 Gaussin menetelm\u00e4n polynomimatriiseille yleistetyill\u00e4 rivioperaatioilla voidaan aina muokata polynomimatriisi yl\u00e4kolmiomuotoon. Toinen puolestaan ottaa k\u00e4ytt\u00f6\u00f6n my\u00f6s sarakeoperaatiot. T\u00e4ll\u00f6in voidaan muokata mik\u00e4 tahansa polynomimatriisi sellaiseksi diagonaalimatriisiksi, jonka nollasta eroavat l\u00e4vist\u00e4j\u00e4polynomit ovat perusmuotoisia, ja edellinen jakaa aina seuraavan. Lis\u00e4ksi nollapolynomit voivat esiinty\u00e4 l\u00e4vist\u00e4j\u00e4ll\u00e4 vain siten, ett\u00e4 nollapolynomia seuraava l\u00e4vist\u00e4j\u00e4polynomi on my\u00f6s nollapolynomi. T\u00e4llaista muotoa olevaa polynomimatriisia kutsutaan alkuper\u00e4isen matriisin Smithin normaalimuodoksi. Se on lis\u00e4ksi yksik\u00e4sitteinen, mik\u00e4 on my\u00f6s tarkoituksena osoittaa. Tulos tarkoittaa my\u00f6s sit\u00e4, ett\u00e4 jokainen polynomimatriisi on ekvivalentti Smithin normaalimuotonsa kanssa.\n\nTutkielman toisena osana on esitellyn polynomimatriisien teorian hy\u00f6dynt\u00e4minen kuntakertoimisten matriisien teoriassa. Yhten\u00e4 keskeisimp\u00e4n\u00e4 tavoitteena on m\u00e4\u00e4ritell\u00e4 kuntakertoimisen matriisin karakteristinen polynomi k\u00e4ytt\u00e4m\u00e4tt\u00e4 lainkaan determinanttia. T\u00e4m\u00e4 tapahtuu hy\u00f6dynt\u00e4m\u00e4ll\u00e4 polynomimatriisin yl\u00e4kolmiomuotoa. Vaihtoehtoisena laskutapana esitet\u00e4\u00e4n my\u00f6s polynomirenkaan osam\u00e4\u00e4r\u00e4kuntaa hy\u00f6dynt\u00e4v\u00e4 keino. Toinen t\u00e4m\u00e4n j\u00e4lkimm\u00e4isen osan p\u00e4\u00e4tavoitteista on m\u00e4\u00e4ritell\u00e4 Smithin normaalimuodon avulla kuntakertoimiselle matriisille similaarisuusinvariantit ja osoittaa, ett\u00e4 niist\u00e4 voidaan p\u00e4\u00e4tell\u00e4 matriisin Frobeniuksen ja Jordanin muodot. Teoria pohjautuu lauseeseen, jonka mukaan kuntakertoimiset matriisit A ja B ovat similaariset t\u00e4sm\u00e4lleen silloin, kun polynomimatriisit A-xI ja B-xI ovat ekvivalentit. Toisin sanoen n\u00e4ill\u00e4 polynomimatriiseilla on silloin sama Smithin normaalimuoto.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Antti Lindberg (antantli) on 2014-08-25 13:22:51.185408. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2014-08-25T13:22:51Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201408252627.pdf: 483520 bytes, checksum: f77cd03fb7690f9d77c67a296d1ea203 (MD5)\nlicense.html: 4775 bytes, checksum: 28a10d0a03f5d06805f8b16a5868e5f6 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2014-08-25T13:22:51Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201408252627.pdf: 483520 bytes, checksum: f77cd03fb7690f9d77c67a296d1ea203 (MD5)\nlicense.html: 4775 bytes, checksum: 28a10d0a03f5d06805f8b16a5868e5f6 (MD5)\n Previous issue date: 2014", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (58 sivua)", "language": null, "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": "Matriisiteoria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", 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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": "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2014-08-25T13:22:52Z", "language": null, 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