N. Halko, P. G. Martinsson, J. A. Tropp: Finding Structure with Randomness:
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions,
SIAM REVIEW, Vol. 53, No. 2, (2011)217–288
Matrix Analysis for Scientists and Engineers
by Alan J. Laub, SIAM, Philadelphia
Alan J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, Philadelphia, 2005
Terminated in academic year 2023/2024
Applied Algebra
| Type of study | Follow-up Master |
|---|---|
| Language of instruction | Czech |
| Code | 470-4201/01 |
| Abbreviation | AA |
| Course title | Applied Algebra |
| Credits | 4 |
| Coordinating department | Department of Applied Mathematics |
| Course coordinator | prof. RNDr. Zdeněk Dostál, DSc. |
Subject syllabus
• An introduction to matrix decompositions with motivation and applications
• Spectral decomposition of a symmetric matrix
• Applications of the spectral decomposition: matrix functions, convergence of iterative methods, extremal properties of the eigenvalues
• QR decomposition – rank of the matrix, atable solution of linear systems, reflection
• SVD – low rank approximations of a matrix, image deblurring, image compression
• Approximate decompositions of large matrices and related linear algebra
• Tensor approximations – Kronecker product, tensors, tensor SVD, tensor train, image debluring
• Variational principle and least squares
• Total least squares
• Minimization of a quadratic function with equality constraints – KKT, duality, basic algorithms, SVM,
• Analytic geometry with matrix decompositions
• Inverse problems – Tichonov regularization, applications
• Spectral decomposition of a symmetric matrix
• Applications of the spectral decomposition: matrix functions, convergence of iterative methods, extremal properties of the eigenvalues
• QR decomposition – rank of the matrix, atable solution of linear systems, reflection
• SVD – low rank approximations of a matrix, image deblurring, image compression
• Approximate decompositions of large matrices and related linear algebra
• Tensor approximations – Kronecker product, tensors, tensor SVD, tensor train, image debluring
• Variational principle and least squares
• Total least squares
• Minimization of a quadratic function with equality constraints – KKT, duality, basic algorithms, SVM,
• Analytic geometry with matrix decompositions
• Inverse problems – Tichonov regularization, applications
Literature
Advised literature
Tamara G. Kolda, Brett W. Bader. Tensor Decompositions and Applications, SIAM Review, Vol. 51, No. 3, (2009)455–500
Carl D. Meyer, Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000
Dianne P. O'Leary, Scientific Computing with Case Studies, SIAM, Philadelphia 2009
Carl D. Meyer, Matrix analysis and applied linear algebra, SIAM, Philadelphia, 2000
Dianne P. O'Leary, Scientific Computing with Case Studies, SIAM, Philadelphia 2009