Matrix Analysis, 8.0 credits

Matrisanalys, 8.0 hp

6FMAI14

Course level

Third-cycle Education

Description

Contact the examiner if interested.

https://courses.mai.liu.se/FU/6FMAI14/

Current and recently held PhD courses at the Department of Mathematics can be found here: https://liu.se/artikel/doktorandkurser-vid-matematiska-institutionen

Contact

Entry requirements

Linear Algebra, honours course (TATA53) or equivalent (the following topics should be familiar: complex vector spaces, the spectral theorem for Hermitian and normal operators, the singular value decomposition, the Jordan normal form).

Contents

  • Special matrices: Toeplitz, circulant, Vandermonde, Hankel, and Hessenberg matrices.
  • Block matrices: inversion formulas, Schur complement.
  • Real and complex canonical forms.
  • Vector and matrix norms.
  • Eigenvalues: location, inequalities, perturbations, Rayleigh quotients, variational characterization. Hadamard´s inequality.
  • Singular values: inequalities, variational characterization, Schatten and Ky Fan norms.
  • Total least squares. Quadratic minimization with linear constraints.
  • Matrix products: Kronecker, Hadamard and Khatri-Rao products.
  • Matrix equations. Stable matrices.
  • Functions of matrices.
  • Matrices of functions, matrix calculus and differentiation.
  • Multilinear algebra, tensor product, decomposition and approximation of tensors.

Educational methods

Lectures.

Examination

Hand-in assigments and oral presentations.

Grading

Two-grade scale

Course literature

Horn and Johnson: Matrix Analysis (recommended), lecture notes.