Simetrične matrike

diplomsko delo

Andreja Hrašovec (Author), Gorazd Lešnjak (Mentor)

Abstract

Simetrične matrike z realnimi elementi so hermitske, zato so unitarno podobne realnim diagonalnim matrikam. Vse njihove lastne vrednosti so realne, prirejene Jordanove kanonične forme so diagonalne matrike. V diplomskem delu predstavimo odgovor na naravno vprašanje, ali velja kaj podobnega v primeru, ko so elementi simetričnih matrik kompleksna števila, ki niso vsa realna. Osnovno znanje o hermitskih matrikah podamo v prvem delu. Dopolnimo ga s karakterizacijo matrik, ki so podobne svoji adjungirani matriki. Za simetrične kompleksne matrike obstaja Takagijeva faktorizacija, kar dokažemo v drugem delu. Tu podamo tudi odgovor na zgornje vprašanje, ki je nikalen, saj velja, da je vsaka matrika s kompleksnimi elementi podobna kakšni simetrični matriki s kompleksnimi elementi. Delo zaključimo z opisom tistih kompleksnih matrik, ki so unitarno podobne kakšni kompleksni simetrični matriki.

Keywords

matematika;simetrične matrike;hermitske matrike;podobnost;unitarna podobnost;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2010
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[A. Hrašovec]
UDC:	51(043.2)
COBISS:	17611272
Views:	3001
Downloads:	284
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	SYMMETRIC MATRICES
Secondary abstract:	Symmetric matrices with real elements are Hermitian, so they are unitarily similar to real diagonal matrices. All their eigenvalues are real and adjusted Jordan canonical forms are diagonal matrices. In Graduation Thesis we give the answer on the following natural question: Is there anything similar in the case, when the elements of symmetric matrices are complex numbers, not all belonging to the set of real numbers. Basic knowledge about Hermitian matrices is given in the first part. We complete it with characterization of the matrices, which are similar to their adjunct matrix. For symmetric complex matrices there exists Takagi's factorization. We prove this in the second part. Here we give the answer to the question above, which is negative, because every matrix with complex elements is similar to some symmetric matrix with complex elements. We finish our work with the description of those complex matrices, which are unitarily similar to any complex symmetric matrix.
Secondary keywords:	symmetric matrix;Hermitian matrix;similarity;unitary similarity;Takagi's factorization;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	57 f., [2] f. pril.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	18488