Matrix Theory
Code  Completion  Credits  Range  Language 

01TMA  Z  2  0P+2C  Czech 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The subject deals mainly with:
1) similarity of matrices and canonical forms of matrices
2) PerronFrobenius 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. Nonnegative matrices and the PerronFrobenius 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 PerronFrobenius 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:

Key references:
[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 references:
[3] S. Friedland, Matrices  Algebra, Analysis and Applications, World Scientific 2016.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans: