Advanced Matrix Analysis

Course guarantor:

Jiří Velebil

Lecturer:

Martin Křepela

Tutor:

Martin Křepela

Supervisor:

Department of Mathematics

Synopsis:

The course covers advanced topics of linear algebra, in particular matrix factorizations and construction of matrix functions.

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.

Syllabus of lectures:

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.

Syllabus of tutorials:

Study Objective:

Study materials:

1. C. D. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM 2000

2. M. Dont: Maticová analýza, skripta, nakl. ČVUT 2011

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue	roomT2:C3-132 Křepela M. 14:30–16:00 (lecture parallel1) Dejvice
Wed	roomT2:C3-132 Křepela M. 16:15–17:00 (lecture parallel1) Dejvice roomT2:C3-132 Křepela M. 17:00–17:45 (lecture parallel1 parallel nr.101) Dejvice
Thu
Fri

The course is a part of the following study plans:

• Open Electronic Systems (compulsory course in the program)
• Open Electronic Systems (compulsory course in the program)