Matrix computations

Teacher(s)

Jungers Raphaël;

Language

English

Prerequisites

Basic knowledge (1st cycle) in linear algebra and numerical analysis.

Main themes

The course builds on the solid mathematical foundations of Matrix theory in order to elaborate algorithmic solutions to major challenges involving computations with/on matrices.

• Matrices defined over a field/ring/nonnegative: applications and challenges
• Canonical forms, decompositions, eigen- and singular values
• Norms, convexity, structured matrices: sparse/adjacency matrices
• Recent computational challenges: Nonnegative Matrix Factorization, matrix semigroups

Learning outcomes

At the end of this learning unit, the student is able to :
1	Contribution of the course to the program objectives : AA1.1, AA1.2 AA5.5 AA6.3 After successful completion of this course, the student will : have acquired a solid basis of matrix theory and its applications in several engineering disciplines understand the use of matrix properties in the solution of these problems have acquired a solid background in matrix problems involving eigenvalues, singular values, non-negative and polynomial matrices have shown how to apply his theoretical background in concrete matrix problems. be able to model an engineering problem by choosing the adequate concepts and the good tool within the wide panel offered by Matrix theory.

Content

After an introduction recalling some basic notions, we discuss the following topics:

1. Complements on determinants
2. The singular value decomposition and its applications. Angles between subspaces, generalized inverses, projectors, least-squares problems
3. Eigenvalue decomposition: Schur and Jordan form
4. Approximations and variational characterization of eigenvalues
5. Congruence and stability: inertia, Lyapunov equation, stability analysis of dynamical systems
6. Structured and Polynomial matrices: Euclid algorithm, Smith normal form, fast algorithms.
7. Nonnegative matrices: Perron-Frobenius theorem, stochastic matrices
8. Matrix semigroups: algebraic structure, algorithms and applications (NMF, Joint Spectral Characteristics)

Teaching methods

• Regular classes with a schedule fixed by the EPL.
• A seminar with presentations by the students is organized at the end of the quadrimester.
• Exercises or homeworks made individually or in small groups, with the possibility to consult teaching assistants..
• Details announced during the first class.

Some activities could be organized in remote fashion, e.g. on MS Teams.

Evaluation methods

The evaluation of the students is partly based on an exam organized according to the rules imposed by the EPL. The exam material corresponds to the contents of the course material,  with the possible exception of certain parts specified after the last session of the course.

The other part of the evaluation is based on the homeworks and presentations made during the semester. This evaluation will only be taken into account during the first session. If there is a second session, the note will only take into account the second exam.

More elaborate information on the evaluation procedure is given in the course plan, made available at the beginning of the academic year.   
For a written exam, in case of doubt, the teacher might invite the student for a supplementary oral exam.

Online resources

http://moodleucl.uclouvain.be/course/view.php?id=7969

Bibliography

Le support de cours se compose d'ouvrages de référence, de notes de cours détaillées et de documents complémentaires disponibles sur Moodle.

Ouvrages de référence :

• G.H. Golub and C.F. Van Loan (1989). Matrix Computations, 2nd Ed, Johns Hopkins University Press, Baltimore.
• P. Lancaster and M. Tismenetsky (1985). The Theory of Matrices, 2nd Ed, Academic Press, New York

Teaching materials

• LINMA 2380 Course notes by R.J. et al.

Faculty or entity

MAP