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master's thesis
Wignerov teorem za slučajne matrice
Zagreb: University of Zagreb, Faculty of Science, 2017. urn:nbn:hr:217:564227
Krklec, Jurica
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

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Krklec, J. (2017). Wignerov teorem za slučajne matrice (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:564227

Krklec, Jurica. "Wignerov teorem za slučajne matrice." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:564227

Krklec, Jurica. "Wignerov teorem za slučajne matrice." Master's thesis, University of Zagreb, Faculty of Science, 2017. https://urn.nsk.hr/urn:nbn:hr:217:564227

Krklec, J. (2017). 'Wignerov teorem za slučajne matrice', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 November 2024, https://urn.nsk.hr/urn:nbn:hr:217:564227

Krklec J. Wignerov teorem za slučajne matrice [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2017 [cited 2024 November 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:564227

J. Krklec, "Wignerov teorem za slučajne matrice", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2017. Available at: https://urn.nsk.hr/urn:nbn:hr:217:564227

Metadata
TitleWignerov teorem za slučajne matrice
AuthorJurica Krklec
MentorVjekoslav Kovač (mentor)
Committee memberVjekoslav Kovač (predsjednik povjerenstva)
Committee memberMirko Primc (član povjerenstva)
Committee memberMarcela Hanzer (član povjerenstva)
Committee memberBojan Basrak (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2017-09-26, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Slučajne matrice su koncept iz teorije vjerojatnosti, koji je motiviran numeričkom linearnom algebrom, teorijskom fizikom, teorijom brojeva, matematičkom statistikom i još mnogim drugim granama matematike i fizike. U ovom radu obrađuju se tzv. realne Wignerove slučajne matrice, koje su simetrične matrice s međusobno nezavisnim jednako distribuiranim elementima na dijagonali i u gornjem trokutu. Ako su \(\lambda_1 \le \lambda_2 \le \cdots \le \lambda_N\) svojstvene vrijednosti takve matrice, promatraju se slučajne vjerojatnosne mjere \(\frac{1}{N} \sum_{i=1}^{N} \delta_{\lambda_i}\). Jedan od najklasičnijih rezultata o slučajnim matricama je Wignerov teorem, koji govori da niz takvih mjera konvergira slabo po vjerojatnosti kada \(N \to \infty\) prema polukružnoj razdiobi, tj. distribuciji s gustoćom \(\sigma(x) = \frac{1}{2 \pi} \sqrt{4-x^2}\) za \(-2 \le x \le 2\). Glavni cilj ovog rada je dati elementarni dokaz tog važnog teorema.
Abstract (english)
Random matrix is a term from probability theory, inspired by numerical linear algrebra, theoretical physics, number theory, mathematical statistics, and many other fields of mathematics and physics. In this thesis we consider the so-called Wigner random matrices, which are symmetric matrices with independent identically distributed diagonal elements and independent identically distributed upper triangular elements. Assuming that \(\lambda_1 \le \lambda_2 \le \cdots \le \lambda_N\) are eigenvalues of such random matrix, we consider a probability measure \(\frac{1}{N} \sum_{i=1}^{N} \delta_{\lambda_i}\). Wigner's semicircle law states that this probability measure converges weakly in probability, as \(N \to \infty\), to the semicircle distribution with density \(\sigma(x) = \frac{1}{2 \pi} \sqrt{4-x^2}\) for \(-2 \le x \le 2\). The main topic of this thesis is an elementary proof of this important theorem.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:564227
Study programmeTitle: Mathematical Statistics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2018-02-14 10:29:17