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Professor: Christian Böhning
Term: 2
Status: Unusual
Assessment: 85% exam, 15% assignmentsRating
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| Topic | Rating | Summary |
|---|---|---|
| Jordan Canonical Form | Abstractly, this solves the classification problem for pairs (V, T) where V is a finite dimensional vector space over the complex numbers (or any other algebraically closed field) and T a linear self-map of V, up to the equivalence relation induced by bijective linear self-maps of V; more concretely, we classify n by n complex matrices A up to conjugation by invertible matrices P, i.e., the operation A ⭢ P^{-1}AP. | |
| Bilinear, sesquilinear and quadratic forms on finite dimensional vector spaces | These structures are ubiquitous and fundamental in mathematics and many parts of the sciences. For example, the standard scalar product in R^n is an example. In passing we mention that the description of amplitudes, probabilities and expectation values in quantum theory places such structures at the very heart of how nature works at the smallest levels. We will cover orthonormal basis, Gram-Schmidt process, diagonalisation, singular value decomposition, hermitian forms and normal matrices, among other things. | |
| Duality | ||
| Tensor, exterior and symmetric algebras |
- P M Cohn, Algebra, Vol. 1, Wiley, 1982
- I N Herstein, Topics in Algebra, Wiley, 1975
- Jorg Liesen and Volker Mehrmann, Linear Algebra, Springer, 2015
- Peter Petersen, Linear Algebra, Springer, 2012
- F. Gantmacher, The Theory of Matrices, American Mathematical Society, 2001
- Peter Lax, Linear Algebra and Its Applications, 2nd Edition, Wiley, 2007