Matrix Theory

Course guarantor:

Edita Pelantová

Lecturer:

Tutor:

Edita Pelantová

Supervisor:

Department of Mathematics

Synopsis:

The subject deals mainly with:

1) similarity of matrices and canonical forms of matrices

2) Perron-Frobenius theory and its applications

3) tensor product

4) Hermitian and positive semidefinite matrices

Requirements:

Successful completion of courses Linear algbera and General algebra.

Syllabus of lectures:

1. The Jordan Theorem and transformation of matrix into its canonical form, invariant subspaces.

2. Canonical forms of matrices with real resp. rational entries.

3. Relation between matrices and graphs

4 Non-negative matrices and the Perron-Frobenius theorem, stochastic matrices.

5. The tensor product of matrices and its properties.

6. Hermitian matrices, the interlacing theorem

7. Positive semidefinite matrices, the Hadamard inequality

Syllabus of tutorials:

Study Objective:

Acquired knowledge: fundamental results in the theory of canonical forms of matrices, in the Perron-Frobenius theory for nonnegative matrices, the spectral theory for the hermitian matrices and the tensor product of matrices.

Acquired skills: applications of these results in the graph theory, for group representations, in the algebraic number theory, in numerical analysis.

Study materials:

Obligatory:

[1] Fuzhen Zhang: Matric Theory, Springer 2011

[2] M. Fiedler, Special Matrices and Their Applications in Numerical Mathematics. Second Edition. Dover Publications, Inc., Mineola, U.S.A., 2008.

Optional:

[3] Shmuel Friedland, Matrices - algebra, analysis and applications, World Scientific 2016.

Note:

Time-table for winter 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
Wed
Thu	roomTR:212 Pelantová E. 10:00–11:50 (parallel nr.101) Trojanova 13
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Matematické inženýrství (compulsory elective course)
• Matematická informatika (compulsory course in the program)