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Selected games from matrix calculus and their applications
BARTUŠKOVÁ, Zuzana
This bachelor's thesis deals with selected parts of matrix calculus and their applications. In the first part of the thesis, basic terms and operations with matrices are described, including their definitions, properties, rank and operations such as addition, multiplication and transposition of matrices. The second part focuses on determinants and inverse matrices, where the methods of their calculation and practical use are discussed. In addition, attention is paid to the linear dependence and independence of vectors, as well as the problem of eigenvalues and eigenvectors, including their calculations. In the final part, the applications of matrix calculus in various areas of mathematics and technology, such as solving systems of linear equations and vector transformations, are presented. The thesis aims to present not only the theoretical foundations of matrix calculus, but also to demonstrate their practical applications in the real world.
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Numerical approximation of matrix eigenvalues
Pres, Jiří ; Zatočilová, Jitka (referee) ; Tomášek, Petr (advisor)
This work deals with numerical methods for the approximation of matrix eigenvalues, which is~a~key problem in many fields of science and technology. Main goal of the thesis is to get to know with several basic algorithms, such as power methods and the QR decomposition. Further to show these algorithms on examples and to compare achieved results. All described algorithms are processed in MATLAB scripts and attached in the appendix. After testing described methods, summary was created regarding their effective usage.
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Solving methods of systems of linear equations
Šotolová, Petra ; Jančařík, Antonín (advisor) ; Novotná, Jarmila (referee)
My bachelor's thesis is focused on methods for solving systems of linear equations. Its aim is to explain the application of these methods for solving systems of linear equations and to assign advantages and the type of system for which each method is advantageous. The thesis should be suitable for elementary and secondary school students, university students, as well as individuals not studying mathematics. It describes substitution method, addition method, comparison method, graphical method, Gaussian elimination, Gauss-Jordan elimination, inverse matrix and Cramer's rule and their application in solving systems of two or more linear equations with two or more unknowns. It includes a general explanation and a specific procedure with examples, advantages and disadvantages of each method, and types of systems for which the method would be used. The thesis examines the number of solutions of systems of equations depending on the form of the system. It also explains basic concepts and provides simplified definitions of the terms used.
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Analytic representation of affine transformations in the plane
Horáčková, Blanka ; Beran, Filip (advisor) ; Zamboj, Michal (referee)
The thesis deals with three main topics - isometric, similar, and affine transformations of the plane from the point of view of analytic geometry. In the first chapter, the most basic concepts are recalled, which will be subsequently dealt with throughout the thesis. The second chapter focuses on identical transformations. Here, we find an analytic representation of this transformation in matrix and complex forms. The third chapter focuses on similar transformations. The central point is then the decomposition of similarity into identity and identicalness, where knowledge of both similarities and identities are combined. The last chapter focuses on affinities. This chapter is not so theoretical anymore but focuses mainly on the characteristic elements of affine transformations and examples. This work's important and beneficial factor is the solved examples, supplemented by several figures. The work is intended primarily for mathematics students as a study material. However, it may also be used by secondary school teachers to supplement the secondary school curriculum.
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Computation of the eigenvalues and eigenvectors of Hermitian matrix
Štrympl, Martin ; Bohrn, Marek (referee) ; Fujcik, Lukáš (advisor)
This project deals with computation of eigenvalues and eigenvectors of Hermitian positive-semidefinite complex square matrix of order 4. The target is an implementation of computation in language VHDL to field-programmable gate array of type Xilinx Zynq-7000. This master project deals with algorithms used for computation of eigenvalues and eigenvectors of positive-semidefinite symmetric real square and positive-semidefinite complex Hermitian matrix and the analysis of algorithms by AnalyzeAlgorithm program assembled for this purpose. The closing part of this project describes implementation of the computation into field-programmable gate array with use of IP core Xilinx® Floating-Point \linebreak Operator and SVAOptimalizer, SVAInterpreter and SVAToDSPCompiler programs.
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Selected Extensions of the Albegraic System Octave
Salač, Radek ; Smrčka, Aleš (referee) ; Vojnar, Tomáš (advisor)
This work deals with issues linked to solving system of linear equations in the environment of numerical computer. It describes the fundamental algorithms emphasizing their positive as well as negative sides. The work is devoted to general issues such as time complexity and memory demandingness of given algorithms. In the last part, the process of implementation of selected procedures into the algebraic system Octave is described.
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Interactive Manipulation with 3D Objects with Force Feedbeck
Bělín, Jan ; Chudý, Peter (referee) ; Kršek, Přemysl (advisor)
Physical haptic interaction is added to the modern manipulation with objects in virtual space. In content of this master's thesis the haptic technology is represented by SensAble Phantom Omni device and OpenHaptics toolkit, which is related to the device. Reader is initially introduced into mathematical basics of manipulation and into haptic technology history including current state. The introduction into Openhaptics toolkit follows as well as HDAPI and HLAPI libraries description. As a result of this theoretical basics demo aplications have been created, that show basic and advanced abilities of the Phantom Omni device. Demos represent the functionality of the device as examples integrating well-known elementary physical laws and events.
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