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[{"key": "dc.contributor.advisor", "value": "Lehtonen, Ari", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Artemenko, Maryia", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-05-24T09:39:55Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-05-24T09:39:55Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/75886", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 matematiikan pro gradu -tutkielma k\u00e4sittelee matriisin Jordanin normaalimuotoa. Jordanin muoto on matriisin muoto, joka on l\u00e4hemp\u00e4n\u00e4 diagonaalimuotoa. Se on hy\u00f6dyllinen tapauksessa, kun matriisi ei ole diagonalisoituva. Matriisin Jordanin muoto voidaan saada v\u00e4hint\u00e4\u00e4n kaksi eri tavalla. T\u00e4ss\u00e4 ty\u00f6ss\u00e4 pohditaan matriisin Jordanin muotoa k\u00e4ytt\u00e4m\u00e4ll\u00e4 pohjimmillaan algebrallisia k\u00e4sitteit\u00e4 ja keinoja. Keskeinen ty\u00f6n k\u00e4site on polynomimatriisi. Polynomimatriisin peruslaskutoimitukset ja alkeismuunnokset voidaan m\u00e4\u00e4ritell\u00e4 vastaavasti kuin tavallisille matriisille. Lis\u00e4ksi jokainen polynomimatriisi ekvivalentti diagonaalimatriisille, jonka l\u00e4vist\u00e4j\u00e4alkiot ovat polynomimatriisin invariantit polynomit. T\u00e4m\u00e4 diagonaalipolynomimatriisi kutsutaan polynomimatriisin Smithin normaalimuodoksi. Matriisin Jordanin muodon rakentamisessa k\u00e4ytet\u00e4\u00e4n alkeistekij\u00e4n k\u00e4site. Alkeistekij\u00e4t l\u00f6ytyy aina kun invarianttien polynomien kertoimet ovat algebrallisesti suljetun kunnan alkioita. T\u00e4m\u00e4n kuntan esimerkkin\u00e4 voi olla kompleksilukujen joukko. \n\n Tutkielma jakaantuu kolmeen osaan. Ensimm\u00e4inen osa sis\u00e4lt\u00e4\u00e4 lineaarialgebran perusk\u00e4sitteit\u00e4 sek\u00e4 polynomin k\u00e4site ja polynomien jakoalgoritmi. Toinen osa sis\u00e4lt\u00e4\u00e4 matriisiteorian perusk\u00e4sitteit\u00e4 ja esitietoja kuten matriisin m\u00e4\u00e4ritelm\u00e4 ja tyyppej\u00e4, laskutoimituksia matriisien v\u00e4lill\u00e4, matriisin determinantti, ominaisarvoteorian perusk\u00e4sitteit\u00e4 jne. Tutkielman kolmannessa osassa m\u00e4\u00e4ritell\u00e4\u00e4n matriisin Jordanin muoto ja Jordan hajotelman muodostaminen alkeistekij\u00f6iden avulla. Jokainen tutkielman luku sis\u00e4lt\u00e4\u00e4 lukuun kuuluvia esimerkkej\u00e4 ja niiden ratkaisut.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "This mathematics master\u2019s thesis is covering the Jordan normal form of a matrix. Jordan normal form is close to the diagonal form. It is useful when a matrix is not diagonalisable. Jordan form of a matrix can be attained at least in two different ways. In this paper, the Jordan normal form is derived using algebraic theory. The main concept of this work is the polynomial matrix. The elementary transformations for a polynomial matrix are the same as for any other matrix. In addition to that, every polynomial matrix is equivalent to a diagonal matrix whose diagonal elements are the invariant polynomials of a polynomial matrix. This kind of diagonal polynomial matrix is called the Smith normal form. The elementary factor is introduced in order to construct the Jordan normal form. The elementary factor can always be found if the coefficients of the invariant polynomial are the elements of an algebraically closed field. It can be for example the set of complex numbers. \n\nThe thesis consists of three parts. The first part contains the basics of linear algebra, polynomials and the algorithm for the polynomial division. The second parts consists of the basics of the matrix theory, such as the matrix definition and types, operations between matrices, matrix determinant, basic concepts of the eigenvalue theory, etc. The Jordan normal form and the Jordan decomposition using elementary factors is defined in the third part. Each chapter of the thesis contains examples and solutions.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-05-24T09:39:55Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-05-24T09:39:55Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "78", "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": "Jordanin muoto", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "polynomimatriisi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "minimipolynomi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "invariantit polynomit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "alkeistekij\u00e4t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matriisin Jordanin muoto", "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-202105243145", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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": "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", 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