UCL - Etudes

Formations
Premier cycle
Deuxième cycle
Troisième cycle
Certificats (programmes non académiques)
Passerelles
Formation continue
Facultés et entités
Cadre académique
Réforme de Bologne
Accès aux études
Organisation des études
Lexique
Calendrier académique
Règlement des études et examens
Charte pédagogique
Renseignements généraux

MATRIX THEORY [INMA2380]
[30h+22.5h exercises] 5 credits

Version française

Printable version

This course is taught in the 2nd semester

Teacher(s):

Paul Van Dooren

Language:

french

Level:

2nd cycle course

>> Aims
>> Main themes
>> Content and teaching methods
>> Other information (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)
>> Other credits in programs

Aims

In-depth study of some specific topics of matrix theory, with emphasis on applications and on underlying numerical aspects.

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.

Content and teaching methods

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 (prerequisite, evaluation (assessment methods), course materials recommended readings, ...)

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

Other credits in programs

INFO22

Deuxième année du programme conduisant au grade d'ingénieur civil informaticien

(5 credits)

MAP21

Première année du programme conduisant au grade d'ingénieur civil en mathématiques appliquées

(5 credits)

MAP22

Deuxième année du programme conduisant au grade d'ingénieur civil en mathématiques appliquées

(5 credits)

Mandatory

MAP23

Troisième année du programme conduisant au grade d'ingénieur civil en mathématiques appliquées

(5 credits)

MATH21/G

Première licence en sciences mathématiques (Général)

(5 credits)

MATH21/S

Première licence en sciences mathématiques (Statistique)

(5 credits)

Mandatory



Ce site a été conçu en collaboration avec ADCP, ADEF, CIO et SGSI
Responsable : Jean-Louis Marchand - Contact : secretaire@fsa.ucl.ac.be
Dernière mise à jour : 25/05/2005