Evklidsko razdaljne matrike velikosti 3 x 3 : delo diplomskega seminarja

Marinko, Anže (author), Jaklič, Gašper (mentor)

Matrika je evklidsko razdaljna (Euclidian distance matrix -- EDM), če obstajajo točke, tako da so elementi matrike kvadrati evklidskih razdalj med temi točkami. V tem delu dokažemo nekaj pomembnih lastnosti EDM. Nato pa se osredotočimo na inverzni problem lastnih vrednosti za EDM. Inverzni problem lastnih vrednosti (inverse eigenvalue problem -- IEP) je sledeč: konstruirati (ali dokazati obstoj) matrike z danim spektrom in določenimi lastnostmi (konkretno, da je matrika EDM). Dobro je znano, da ima IEP za EDM velikosti 3x3 rešitev. V tem delu so podane vse rešitve tega problema, proučujemo njihovo povezavo z geometrijo ter možno razširitev na večje EDM z uporabo obrobljenih matrik. Poleg tega pa predstavimo tudi povezavo med znanim problemom obstoja Hadamardovih matrik in IEP za EDM.

• Language: Slovenian
• Keywords: Evklidsko razdaljne matrike, inverzni problem lastnih vrednosti, obrobljene matrike, lastne vrednosti, Hadamardove matrike
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103262
• UDC: 512
• COBISS.SI-ID: 18435417
• Publication date in RUL: 15.09.2018

Secondary language

• Language: English
• Title: Euclidean distance matrices of size 3

Abstract

The matrix is an Euclidean distance (EDM) if there exist points so that the matrix elements are squares of the euclidean distances between these points. In this work, we prove some important properties of EDM. Then we focus on the inverse eigenvalue problem for EDM. The inverse eigenvalue problem is as follows: to construct (or to prove the existence of) a matrix with a given spectrum and required properties (in particular that the matrix is EDM). It is well known that the IEP for EDM of size 3 has a solution. Here we find all the solutions to this problem, we study their connection with geometry and possible extension to larger EDM using bordered matrices. Then we show the connection between the well known problem of the existence of Hadamard matrices and the IEP for EDM.

• Keywords: Euclidean distance matrices, inverse eigenvalue problem, bordered matrices, eigenvalues, Hadamard matrices