Matrix Theory

Course guarantor:

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:

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 time-table has been prepared for this course

The course is a part of the following study plans: