Advanced Matrix Analysis

Course guarantor:

Jiří Velebil

Lecturer:

Martin Křepela, Jakub Rondoš

Tutor:

Martin Křepela, Jakub Rondoš

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. Review of the basics of linear algebra, norms, matrix norms

2. Scalar products, projections and orthogonal projections, Gram-Schmidt orthogonalization process, least squares method, QR-decomposition

3. Unitary and orthogonal matrices, Householder method

4. Eigenvalues, eigenvectors, and eigensubspaces, Gershgorin circles, diagonalization

5. Schur decomposition, spectral decomposition of diagonalizable matrices, normal and Hermitian matrices, unitary diagonalization

6. Positively definite and semidefinite matrices, singular decomposition, Cholesky decomposition

7. Matrix index, nilpotent matrix, CND decomposition

8. Jordan canonical form of a matrix, spectral projection

9. Construction of a matrix function using power series and spectral decomposition theorem

10. Representation of a matrix function using the Hermite interpolation polynomial, Vandermonde system

11. Matrix exponential, solution of systems of linear ODEs with constant coefficients

Possible extensions:

LU factorization, numerical stability of GEM

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:

Further information:

https://math.fel.cvut.cz/en/people/rondojak/maticovy-pocet

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

	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 Rondoš J. Křepela M. 14:30–16:00 (lecture parallel1) Dejvice
Wed	roomT2:C3-132 Rondoš J. Křepela M. 16:15–17:00 (lecture parallel1) Dejvice roomT2:C3-132 Rondoš J. 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)