Matrix theory [ LINMA2380 ]
5.0 crédits ECTS
30.0 h + 22.5 h
1q
| Teacher(s) |
Van Dooren Paul ;
|
| Language |
French
|
Place
of the course |
Louvain-la-Neuve
|
| Main themes |
- Matrices defined over a field: equivalence classes, Gaussian elimination, Hermitian forms. similarity and related questions (Courant-Fischer theorem, Schur lemma, QR algorithm, matrix functions, etc.), determinants (Binet-Cauchy theorem), generalized inverses and singular value decomposition with applications
- Matrices defined over a ring: Euclid's algorithm and applications in polynomial matrices, relation to the canonical forms of Hermite and Smith
- Norms and convexity: theory and applications of non-negative matrices, localization of eigenvalues
- Structured matrices: complexity of fast algorithms.
|
| Aims |
In-depth study of some specific topics of matrix theory, with emphasis on applications and on underlying numerical aspects.
|
| Content |
After an introduction recalling some basic notions, we discuss the following topics:
1. Complements on determinants: theorems of Binet-Cauchy and Laplace
2. The singular value decomposition and its applications : polar decomposition, angles between subspaces, generalized inverses, projectors, least-squares problems, regularization
3. Eigenvalue decomposition: Schur and Weyr forms, Jordan form, QR algorithm
4. Approximations and variational characterization: Courant-Fischer and Wielandt-Hoffmann theorem, field of values and Gershgorin theorem
5. Congruence and stability: inertia, Sylvester theorem, Stein and Lyapunov equations, link to stability analysis of dynamical systems
6. Polynomial matrices: Euclid algorithm and the Smith and Hermite forms, link to the Jordan form
7. Non-negative matrices: Perron-Frobenius theorem, stochastic matrices.
8. Structured matrices: notion of displacement rank and fast algorithms for Toeplitz and Hankel matrices.
|
| Other information |
Basic knowledge (1st cycle) in linear algebra and numerical analysis
|
Cycle et année
d'étude |
> Master [120] in Mathematical Engineering
> Master [120] in Electrical Engineering
> Master [120] in Electro-mechanical Engineering
> Master [60] in Mathematics
> Master [120] in Physics
> Master [120] in Statistics: General
> Master [120] in Mathematics
|
Faculty or entity
in charge |
> MAP
|
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