Advanced Matrix Analysis

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This is a grouped Moodle course. It consists of several separate courses that share learning materials, assignments, tests etc. Below you can see information about the individual courses that make up this Moodle course.
Advanced Matrix Analysis (Main course) A8B01AMA
Credits 4
Semesters Summer
Completion Assessment + Examination
Language of teaching Czech
Extent of teaching 3P+1S
The course covers advanced topics of linear algebra, in particular matrix factorizations and construction of matrix functions.
Course outlines
Main topics:

1. Inner product, norm, norm equivalence in finite-dimenstional spaces.
2. Projectors and othogonal projectors, Gram-Schmidt orthogonalization method, QR factorization.
3. Unitary and orthogonal matrices, Householder reduction.
4. Singular value decompostition.
5. Eigenvalues, eigenvectors and eigenspaces, diagonalization, Cholesky factorization.
6. Schur decomposition, normal and Hermitian matrices.
7. Matrix index, nilpotent matrices.
8. Jordan form of a matrix, spectral projectors.
9. Construction of a matrix function by power series and through the spectral decomposition theorem.
10. Matrix functions as Hermite polynomials, Vandermonde system.
11. Matrix exponential, solutions to systems of linear ODE with constant coefficients.

Possible extenstions:

LU factorization, numerical stability of GEM, least squares.
1. C. D. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM 2000
2. M. Dont: Maticová analýza, skripta, nakl. ČVUT 2011
Good knowledge of fundamental topics of linear algebra and single-variable analysis is a prerequisity. Some of the course topics need implemenation of multivariable analysis concepts (normed spaces, power series). It is thus recommended to complete a multivariable analysis course (MA2) before registering for this course.
Matrix Calculus XP01MTP
Credits 4
Semesters Summer
Completion Exam
Language of teaching Czech
Extent of teaching 2P+1S
Similar matrices. Jordan blocks, Jordan canonical matrices. Real canonical form of a real matrix. Characteristic and minimal polynomial. Caley-Hamilton thoerem. Functions of matrices, exponential matrix. Symetric, orthogonal and positive matrices. Diagonalization of symetric, positive and circulant matrices. Singular value decomposition. Moore-Penrose pseudoinverse matrix. Generalized solution of systems of linear equations.
1. F. Zhang: Matrix Theory, Basic Results and Techniques. Springer, 1999
2. D. S. Bernstein: Matrix Mathematics: Facts, and Formulas with Application to Linear Systems. Princeton Univ. Press, 2005
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