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undergraduate thesis
Spektralna dekompozicija i primjene
2020. urn:nbn:hr:126:621418
Valjetić, Mihailo
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Applied Mathematics and Computer Science
Computer Science and Machine Learning Research Group

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Valjetić, M. (2020). Spektralna dekompozicija i primjene (Undergraduate thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:621418

Valjetić, Mihailo. "Spektralna dekompozicija i primjene." Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:621418

Valjetić, Mihailo. "Spektralna dekompozicija i primjene." Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:621418

Valjetić, M. (2020). 'Spektralna dekompozicija i primjene', Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 28 January 2025, https://urn.nsk.hr/urn:nbn:hr:126:621418

Valjetić M. Spektralna dekompozicija i primjene [Undergraduate thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2020 [cited 2025 January 28] Available at: https://urn.nsk.hr/urn:nbn:hr:126:621418

M. Valjetić, "Spektralna dekompozicija i primjene", Undergraduate thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:126:621418

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Metadata
TitleSpektralna dekompozicija i primjene
Title (english)Spectral Decomposition and Applications
AuthorMihailo Valjetić
MentorSnježana Majstorović (mentor)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Applied Mathematics and Computer Science)
(Computer Science and Machine Learning Research Group)
Osijek
Defense date and country2020-07-20, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Applied Mathematics and Mathematical Modeling
Abstract
U ovom radu prezentiramo teorijsku pozadinu iza spektralne dekompozicije i neke njene primjene. Najprije se fokusiramo na ulogu svojstvenih vrijednosti i svojstvenih vektora u problemu dijagonalizacije linearnih operatora postupno uvodeći glavne pojmove vezane za svojstveni problem. Zatim promatramo svojstva spektra nekih specijalnih linearnih operatora koji djeluju na unitarnim prostorima predočavajući glavne rezultate popraćene primjerima i adekvatnom geometrijskom intuicijom. Od primjena spektralne dekompozicije ističemo Teorem o glavnim osima i njegovu uporabu u klasifikaciji krivulja drugog reda. Generaliziranjem te tvrdnje dolazimo do primjene naslovne dekompozicije u analizi glavnih komponenti koja ima izrazito veliku uporabu u statistici i nenadziranom strojnom učenju. Za kraj, predstavljamo problem spektralnog particioniranja grafa u teoriji grafova korištenjem minimalnog reza.
Abstract (english)
This paper will present the theoretical background for spectral decomposition and some of its applications. Firstly, we will focus on the role of eigenvalues and eigenvectors in the diagonalization of linear operators by gradually introducing key concepts related to eigendecomposition. Then we observe the spectral properties of some special linear operators defined on unitary spaces by providing main results supported by examples and appropriate geometric intuition. When it comes to the applications of spectral decomposition, we will mention the Principal axes theorem and its relevance in classification of conic sections in the plane. By generalizing this theorem, we will introduce the Principal component analysis encountered in statistics and unsupervised machine learning. In the end, we will discuss the application of spectral decomposition in graph theory, specifically min-cut problem.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:621418
Study programmeTitle: University undergraduate study programme in mathematics
Study programme type: university
Study level: undergraduate
Academic / professional title: sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) matematike (sveučilišni/a prvostupnik/prvostupnica (baccalaureus/baccalaurea) matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
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Created on2020-08-24 07:04:57