Subject description - A8B01AMA
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Explanatory Notes
Instructions
Anotation:
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.
Exercises outline:
Literature:
| 1. | | C. D. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM 2000 |
| 2. | | M. Dont: Maticová analýza, skripta, nakl. ČVUT 2011 |
Requirements:
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.
Subject is included into these academic programs:
| Page updated 13.10.2024 09:51:04, semester: L/2023-4, Z/2025-6, Z,L/2024-5, Send comments about the content to the Administrators of the Academic Programs |
Proposal and Realization: I. Halaška (K336), J. Novák (K336) |