# Subject description - A2M01VKM

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A2M01VKM Selected chapters in mathematics
Roles:V Extent of teaching:4+2
Department:13101 Language of teaching:
Guarantors:  Completion:Z,ZK
Lecturers:  Credits:8
Tutors:  Semester:Z

Anotation:

The first part is devoted to some problems of matrix analysis, especially to propertries connected with eigenvalues and eigenvectors of matices. That is, for example, similarity of matrices, spectral decomposition and the singular value decomposition with applications. In the second part notions of partial differential equations and boundary value problems for partial differential equations are explained. Some concrete boundary value problems are considered and solved using Fourier method and using special function, in particular Bessel and Legendre spherical functions.

Course outlines:

Part I - matrix analysis
 1 Basic notions of linear algebra. Matrix algebra, matrix multiplication, block matrices. 2 Scalar multiplication, vector norm, Gramm-Schmidt ortogonalization. 3 Hermitian, unitary and real orthogonal matrices. 4 Eigenvalues and eigenvectors of a matrix. Similar matrices, diagonalizable matrices. 5 Spectral theorem for hermitian matrices. Schur triangularization theorem and normal matrices. Definite matrices. 6 Discrete Fourier transform, Fourier matrix, fast Fourier transform. 7 Singular value decomposition, least squares problem, Moore-Penrose pseudoinverse.
Part II - PDE and the Fourier method
 8 The notion of PDE, equations of the second kind, wave equation,heat equation, Laplace an Poisson equations. Boundary value problems. 9 Vibrating string, vibrating rectangular membrane, the Fourier method. 10 Wave equation in cylindrical coordinates, Bessel equation and Bessel functions. 11 Fourier-Bessel and Dini-Bessel series. Solution of the problem of vibrating circular membrane. 12 Problems with spherical symmetry, Legendre equation, Legendre spherical functions. 13 Solutions of some boundary value problems using spherical
functions.

Exercises outline:

Part I 1. Spaces R^n and C^n, linear subspaces, bases and dimension.
 2 Scalar multiplication, Gramm-Schmidt ortogonalization. 3 Matrix multiplication, block matrix multiplication, systems of linear equations. 4 Eigenvalues and eigenvectors of matrices. 5 Diagonalizable matrices. 6 Unitary and orthogonal diagonalization of hermitian and real symmetric matrices. 7 Positive definite and semidefinite matrices. 8 Singular value decomposition and the least squares problem.
Part II
 9 PDE and some boundary value problems. 10 Fourier method for the vibrating string problem and for the one-dimensional heat equation. 11 Vibrations of rectangular membrane and the double Fourier series. 12 Vibrations of circular membrane, Bessel functions. 13 Boundary value problems with spherical symmetry and spherical functions.

Literature:

 1 PDF soubory kap1 až kap4 dostupné pomocí ftp na math.feld.cvut.cz, adresář pub/dont/2009 (texty z teorie matic, česky). 2 D. C. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM 2000. 3 M. Dont: Úvod do parciálních diferenciálních rovnic, Nakl. ČVUT, druhé přeprac. vydání 2008. 4 E. A. Gonzáles-Velasco: Fourier Analysis and Boundary Value Problems, Academic Press 1995.

Requirements:

Požadavkem k udělení zápočtu je aktivní účast na cvičení. Požadavky ke zkoušce budou upřesněny během přednášky.

Subject is included into these academic programs:

 Program Branch Role Recommended semester MPIB Common courses V –

 Page updated 22.1.2021 09:51:56, semester: Z/2020-1, L/2021-2, L/2020-1, Z/2021-2, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)