An advanced course on matrix theory and its applications in science and engineering. This course covers most of the important matrix decompositions and their use in solving particular science and engineering problems; it also pays attention to the algorithmic aspects of these decompositions.

Contents :

  • Matrices defined over a field: equivalence classes, Gaussian elimination, determinants, generalized inverses and singular value decomposition with applications
  • Canonical forms under general or orthogonal similarity transformations
  • Localization and perturbations of eigenvalues, inertia of a matrix
  • Matrices defined over a ring: Euclid's algorithm and applications in polynomial matrices, relation to the Smith canonical form
  • Non-negative matrices and the theory of Perron-Frobenius
  • Structured matrices: complexity of fast algorithms for Toeplitz and Hankel matrices