Aims
Indepth 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 (CourantFischer theorem, Schur lemma, QR algorithm, matrix functions, etc.), determinants (BinetCauchy 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 nonnegative 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 BinetCauchy and Laplace
2. The singular value decomposition and its applications : polar decomposition, angles between subspaces, generalized inverses, projectors, leastsquares problems, regularization
3. Eigenvalue decomposition: Schur and Weyr forms, Jordan form, QR algorithm
4. Approximations and variational characterization: CourantFischer and WielandtHoffmann 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. Nonnegative matrices: PerronFrobenius 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
MAP22

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

(5 credits)

Mandatory

