| fullrecord |
[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Carlson, Sanni", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2015-02-11T07:40:30Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2015-02-11T07:40:30Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2015", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1466081", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/45290", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n vektori- ja matriisinormeja, niiden ominaisuuksia\r\nja niihin liittyvi\u00e4 tuloksia. Matriisinormien tarkastelemiseksi on ensin mielek\u00e4st\u00e4 tiet\u00e4\u00e4, \r\nmik\u00e4 on vektorinormi ja millaisia ominaisuuksia silt\u00e4 vaaditaan. Vektorinormilla\r\nvoidaan esimerkiksi laskea vektorin pituus. Matriisinormi taas mittaa esimerkiksi sit\u00e4,\r\nkuinka paljon maksimissaan vektori venyy matriisilla kerrottaessa.\r\nVektorinormeille asetetaan kolme vaatimusta, joiden kaikkien tulee olla voimassa:\r\npositiivisuus, homogeenisuus ja kolmioep\u00e4yht\u00e4l\u00f6. Koska matriisit koostuvat vektoreista,\r\nsiirtyv\u00e4t vektorinormien vaatimukset suoraan matriisinormeille. Vektorinormien\r\nvaatimusten lis\u00e4ksi matriisinormeille m\u00e4\u00e4ritell\u00e4\u00e4n viel\u00e4 yksi ehto lis\u00e4\u00e4; matriisitulon\r\nsubmultiplikatiivisuuden tulee olla voimassa. Nelj\u00e4nnen ehdon lis\u00e4\u00e4minen matriisinormeille\r\naiheuttaa sen, ett\u00e4 kaikki vektorinormit eiv\u00e4t ole matriisinormeja.\r\nOn olemassa useita erilaisia vektori- ja matriisinormeja, joita hy\u00f6dynnet\u00e4\u00e4n tapauskohtaisesti.\r\nYleisin tunnettu vektorinormi on euklidinen normi. Matriisinormeista\r\ntutkielman kannalta oleellinen on spektraalinormi, jossa tarvitaan matriisin ominaisarvojen\r\nhallintaa. Joillekin matriisinormeille voidaan m\u00e4\u00e4r\u00e4t\u00e4 yhteensopiva vektorinormi,\r\njolloin sanotaan, ett\u00e4 matriisinormi sopeutuu vektorinormiin. Lis\u00e4ksi jokainen\r\nvektorinormi indusoi matriisinormin. Indusoitu matriisinormi aina my\u00f6s sopeutuu\r\nvektorinormiin, josta se on indusoitu. Esimerkiksi euklidinen normi indusoi\r\nspektraalinormin ja samalla siis spektraalinormi sopeutuu euklidiseen normiin. Kaikki\r\nmatriisinormit eiv\u00e4t ole indusoituja matriisinormeja.\r\nSpektraalis\u00e4de m\u00e4\u00e4ritell\u00e4\u00e4n matriisin itseisarvoltaan suurimmaksi ominaisarvoksi.\r\nSpektraalis\u00e4de ei ole vektori- eik\u00e4 matriisinormi. Kuitenkin voidaan todistaa, ett\u00e4\r\njokin matriisinormi saadaan \u00e4\u00e4rimm\u00e4isen l\u00e4helle spektraalis\u00e4dett\u00e4. Er\u00e4s spektraalis\u00e4teeseen\r\nliittyv\u00e4 t\u00e4rke\u00e4 tulos antaa yhteyden my\u00f6s matriisijonon suppenemiselle: matriisitulo\r\nsuppenee, jos ja vain jos spektraalis\u00e4de on pienempi kuin yksi. Matriisijonon suppeneminen\r\nm\u00e4\u00e4ritell\u00e4\u00e4n alkioittain eli matriisijono suppenee kohti jotakin tietty\u00e4 matriisia,\r\njos kaikki matriisin alkiot suppenevat kohti jotakin tietty\u00e4 alkiota.\r\nMatriisinormeja sovelletaan muun muassa k\u00e4\u00e4nteismatriisien virheiden arvioinnissa.\r\nMatriisille voidaan m\u00e4\u00e4ritt\u00e4\u00e4 ehtoluku, joka mittaa esimerkiksi matriisin alkioissa\r\nmahdollisesti sattuneiden py\u00f6ristysvirheiden suuruutta. Matriisin ehtoluku riippuu\r\nvalitusta matriisinormista. Mit\u00e4 suurempi ehtoluku on, sit\u00e4 suurempi on matriisin\r\nvirhealttius.\r\nT\u00e4m\u00e4n tutkielman lopussa esitet\u00e4\u00e4n opitun teorian pohjalta kurssisuunnitelma\r\n\"Matriisit tutuiksi\", jonka on tarkoitus olla lukion pitk\u00e4n matematiikan syvent\u00e4v\u00e4\r\nkurssi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Sanni Carlson (sarianre) on 2015-02-11 07:40:30.564844. 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 2015-02-11T07:40:30Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201502111286.pdf: 410423 bytes, checksum: c5bc03cdfc24987b5bb459a1e20d3401 (MD5)\r\nlicense.html: 4773 bytes, checksum: 8e4ffea61d3a0bba9352e4e8d952045c (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2015-02-11T07:40:30Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201502111286.pdf: 410423 bytes, checksum: c5bc03cdfc24987b5bb459a1e20d3401 (MD5)\r\nlicense.html: 4773 bytes, checksum: 8e4ffea61d3a0bba9352e4e8d952045c (MD5)\r\n Previous issue date: 2015", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (55 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": "vektorinormi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisinormi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "spektraalis\u00e4de", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ehtoluku", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matriisinormeista", "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-201502111286", "language": null, "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": "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": "2015-02-11T07:40:31Z", "language": "", "element": "date", "qualifier": "updated", "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": "fi", "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": "vektorit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
|