Advanced linear algebra

To supplement the fundamentals of linear algebra for solving problems arising from signal processing or numerical modelling.

The following topics will be covered

* Eigenvalue/singular value decomposition:

- definition, existence, characterisation.

- Low-rank approximation and link to PCA.

- Some elements of perturbation theory (related to data uncertainty and finite arithmetic calculation).

- Main numerical methods for calculating eigenpairs (QR, iterative power and subspace methods, Lanczos/Arnoldi method).

* Solving systems of linear equations

- characterisation of solutions: inverse, least squares and pseudo-inverse

- some elements of perturbation theory - link with conditioning and finite arithmetic

- main numerical methods for solving linear systems (based on factorisation or iterative methods) fixed point and nested subspace methods

- convergence acceleration principle (preconditioning)

basic knowledge of vector spaces and matrix calculus.