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Eigenvalues of Matrices and Their Localization
Borzíková, Žofia ; Škorpilová, Martina (advisor) ; Halas, Zdeněk (referee)
The diploma thesis is concerned with the topic of eigenvalues of matrices and their lo- calization in the complex plane. First introducing general theorems concerning eigenvalues, eigenvalues for special classes of matrices are then discussed. After presenting the theory of Jordan and Weyr canonical forms, the connection and relation of these two forms is also explained. The estimates of the localizations of the eigenvalues follows from Gershgo- rin's theorem. This text might be used as a didactic material for college-level students of mathematics, thanks to its form having theoretical parts accompanied by examples with commented solutions. It may also be used as a source of information for anyone interested in extending their knowledge of linear algebra. 1
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Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Zamboj, Michal (advisor) ; Jančařík, Antonín (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
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Okrajové úlohy pro rovnice 2. řádu se skákajícími nelinearitami
ZAHRADNÍKOVÁ, Michaela
The subject of this Thesis is to examine nontrivial solutions of boundary value problems for second order ODEs with unilateral jumping nonlinearities. Considered problems can be interpreted as models of a simple beam supported by three types of elastic obstacles. Couples of eigenvalues and eigenfunctions are found and discussed with respect to parameter which represents the strength of the obstacle.
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Affine mappings and transformations in the plane with solved examples
Barborka, Lukáš ; Tůmová, Veronika (advisor) ; Zamboj, Michal (referee)
Analytical geometry widely uses the apparatus of linear algebra, it is, of course, its natural application. The aim of this thesis is the theoretical interconnection, for many students still abstract, bases of the linear algebra with their practical application in the analyti- cal geometry, especially in affine transformations and their use in the solved examples in the plane. This thesis is intended to put concepts known from the course of Linear algebra (homomorphism, eigenvalues/eigenvectors, orthogonal matrices, transition matri- ces...) into context with practical using in the analytical geometry, whether in the form of proofs of important theorems using the linear algebra and arithmetic apparatus, or the following solved examples. The aim of the examples is to provide some insight or guidance on the solution of the same or analogous tasks. The theory and examples are in some cases supplemented with illustrations for better clarity. The work is divided into several parts for greater clarity. The introduction is repeated important concepts of linear algebra such as group, field, vector space, Euclidean space, linear mapping (homomorphism), change of coordinates matrix, eigenvalue/eigenvector of the matrix. It also switches to affine point space, affine coordinate system, transformation equation for...
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Methods of Static Buckling Analysis
Svoboda, Filip ; Eliáš, Jan (referee) ; Frantík, Petr (advisor)
The aim of this theses is to create application, which is able to calculate buckling load of structure made from 1D bar elements, using finite element method. introduction is devoted to basic principles of buckling and derivation of necessary formulas. Then are described all operations and numerical methods needed for the application. At the and is in detail analyzed few structures and results are compared with known solutions or with other applications.
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Development of algorithms for digital real time image processing on a DSP Processor
Knapo, Peter ; Sajdl, Ondřej (referee) ; Belgium, Jurgen Baert (MSc), KHBO (advisor)
Rozpoznávanie tvárí je komplexný proces, ktorého hlavným ciežom je rozpoznanie žudskej tváre v obrázku alebo vo video sekvencii. Najčastejšími aplikáciami sú sledovacie a identifikačné systémy. Taktiež je rozpoznávanie tvárí dôležité vo výskume počítačového videnia a umelej inteligencií. Systémy rozpoznávania tvárí sú často založené na analýze obrazu alebo na neurónových sieťach. Táto práca sa zaoberá implementáciou algoritmu založeného na takzvaných „Eigenfaces“ tvárach. „Eigenfaces“ tváre sú výsledkom Analýzy hlavných komponent (Principal Component Analysis - PCA), ktorá extrahuje najdôležitejšie tvárové črty z originálneho obrázku. Táto metóda je založená na riešení lineárnej maticovej rovnice, kde zo známej kovariančnej matice sa počítajú takzvané „eigenvalues“ a „eigenvectors“, v preklade vlastné hodnoty a vlastné vektory. Tvár, ktorá má byť rozpoznaná, sa premietne do takzvaného „eigenspace“ (priestor vlastných hodnôt). Vlastné rozpoznanie je na základe porovnania takýchto tvárí s existujúcou databázou tvárí, ktorá je premietnutá do rovnakého „eigenspace“. Pred procesom rozpoznávania tvárí, musí byť tvár lokalizovaná v obrázku a upravená (normalizácia, kompenzácia svetelných podmienok a odstránenie šumu). Existuje mnoho algoritmov na lokalizáciu tváre, ale v tejto práci je použitý algoritmus lokalizácie tváre na základe farby žudskej pokožky, ktorý je rýchly a postačujúci pre túto aplikáciu. Algoritmy rozpoznávania tváre a lokalizácie tváre sú implementované do DSP procesoru Blackfin ADSP-BF561 od Analog Devices.
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