Course contents *
1. Review of vector spaces, inner product, determinants, rank
2. Eigenvalues, eigenvectors characteristic polynomial
3. Unitary equivalence QR-factorization
4. Canonical forms Jordan form, polynomials and matrices
5. Hermitian and symmetric matrices Variational characterization of eigenvalues, simultaneous diagonalization
6. Norms for vectors and matrices
7. Location and perturbation of eigenvalues
8. Positive definite matrices. Singular value decomposition
9. Nonnegative matrices, positive matrices, stochastic matrices
10. Stable matrices; Lyapunovs theorem
11. Matrix equations and the Kronecker product, Hadamard product
12. Matrices and functions square roots, differentiation
Intended learning outcomes *
The course is given every second year and is suitable as a first postgraduate (Ph.D) course or as an advanced course in the final year of the M.Sc. program (e.g., for students considering to join the PhD program).
We will refresh and extend the basic knowledge in linear algebra from previous courses in the undergraduate program. Matrix algebra is of fundamental importance for scientists and engineers in many disciplines. In this course we will focus on topics that are of particular interest in communications, signal processing and automatic control.
The course requires a large amount of self study and homework problems will be handed out every week and will be due the following week. It assumes some familiarity with basic concepts from linear algebra (as can be expected by good knowledge from undergraduate studies).
After the course, each student is expected to be able to:
- Show a good working knowledge of some fundamental tools (specified by the course content) in matrix algebra.
- Use the acquired knowledge to more easily apprehend research papers in engineering.
- Identify research problems in which matrix algebra tools may be powerful.
- Apply the knowledge to solve the identified matrix algebra problems.
- Combine several sub problems and solutions to solve more complex problems.