Matrix Theory

Garant předmětu:

Lecturer:

Tutor:

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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

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