470-4201/01 – Applied Algebra (AA)
Gurantor department | Department of Applied Mathematics | Credits | 4 |
Subject guarantor | prof. RNDr. Zdeněk Dostál, DSc. | Subject version guarantor | prof. RNDr. Zdeněk Dostál, DSc. |
Study level | undergraduate or graduate | Requirement | Optional |
Year | 1 | Semester | summer |
| | Study language | Czech |
Year of introduction | 2015/2016 | Year of cancellation | |
Intended for the faculties | FEI | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
A sudent will get basic knowledge of linear and multilinear algebra and their applications in modern information technology.
Teaching methods
Lectures
Tutorials
Summary
Vector space, orthogonality, special bases (hierarchical, Fourier, wavelets), linear mapping, bilinear an quadratic forms, matrix decompositions (spectral, Schur, SVD), Markov's precesses, Page Rank vector, linear algebra of huge matrices, low rank approximation of large matrices, quadratic programming, SVM, tensors. Applications in information technology.
Compulsory literature:
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
Recommended 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
Way of continuous check of knowledge in the course of semester
Two tests during the term.
E-learning
Other requirements
There are no additional requirements imposed on the student.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
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
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction