Primjene linearne algebre na magične kvadrate

master's thesis

• Title (english): Applications of linear algebra to magic squares
• Author: Valentina Herceg
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu je prikazano kako se neki od osnovnih pojmova i činjenica iz linearne algebre primjenjuju na magične kvadrate. Magični kvadrat, u smislu kvadratne tablice koja je popunjena brojevima tako da jednaku vrijednost ima zbroj elemenata po svakom retku, po svakom stupcu i po obje dijagonale, ima vrlo dugu povijest kao popularni objekt iz takozvane zabavne matematike, ali također i kao predmet ozbiljnog matematičkog istraživanja. Promatrani kao matrice nad poljem, obično nad realnim brojevima, magični kvadrati jednakog reda \(n\) čine vektorski prostor, a neki posebni tipovi čine njegove potprostore. Izračunata je dimenzija prostora magičnih kvadrata i nekih potprostora. Za tu svrhu, kao i za proučavanje spektra, korisnom se pokazuje jednoznačan rastav magičnog kvadrata konstante \({\mu}\) kao zbroj jednog kvadrata s magičnom konstantom 0 i drugog kojem su svi elementi jednaki \(-\frac{\mu}{n}\). Proučavaju se neki posebni tipovi magičnih kvadrata, čija struktura vodi do zanimljivih rezultata i konstrukcija. Kroz definicije, brojne primjere i dokaze propozicija vidljiva je prikladnost linearne algebre u istraživanju magičnih kvadrata. Dokazano je da je svaki centrosimetrični magični kvadrat parnog reda singularna matrica. S druge strane, za svaki neparni red \(n\), koji je pritom prost broj, može se konstruirati centrosimetrični klasični magični kvadrat koji je regularna matrica.

Abstract (english)

In this thesis it is shown how some basic concepts and facts from linear algebra are applied to magic squares. Magic square, in the sense of a quadratic scheme filled with numbers in such a manner that the same value is obtained by adding up the entries in each row, each column and both diagonals, has a very long history as a popular object in recreational mathematics, but also as a topic of serious mathematical research. Observed as quadratic matrices over a field, usually the real number field, magic squares of order \(n\) constitute a vector space, with subspaces formed by squares of some particular types. The dimension of this space and some of its subspaces is determined. It is shown how a unique decomposition of a square with magic constant \({\mu}\) as the sum of one square with magic constant 0 and another one with all its elements equal to \(-\frac{\mu}{n}\) becomes useful to that purpose, as well as for exploration of the spectrum of a magic square. Some special types of magic squares are studied, as their structure leads to interesting results and constructions. Through the definitions, numerous examples and the proofs of propositions it is visible how suitable linear algebra is for research of magic squares. It is proven that every centrosymmetric magic square of even order is a singular matrix. On the other hand, for any given odd prime number n it is possible to build a classical, centrosymmetric magic square of order \(n\).

• Keywords: magični kvadrat; vektorski prostor
• Keywords (english): magic squares; vector space
• Language: croatian
• URN:NBN: urn:nbn:hr:217:550832
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-02-25 12:25:14