Introductory description

It is a second Linear Algebra module, where advanced linear algebra concepts are rigorously developed for students familiar with algebraic tools.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

The module is divided in three part:

The first main topic is the Jordan canonical form and related results. 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.

Secondly, we treat bilinear, sesquilinear and quadratic forms on finite dimensional (real and complex) 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.

The third part is concerned with a thorough discussion of the very useful concept of duality (dual vector spaces, dual linear maps, dual bases etc.) and its applications, and after that tensor, exterior and symmetric algebras and their basic properties.

Learning outcomes

By the end of the module, students should be able to:

• develop full command of the theory and computation of the the Jordan canonical form of matrices and linear maps
• learn how to define and to compute functions of matrices
• develop the working knowledge of bilinear forms and quadratic forms
• master the concept of tensor and get proficient manipulating tensors

Indicative reading list

P M Cohn, Algebra, Vol. 1, Wiley, 1982
I N Herstein, Topics in Algebra, Wiley, 1975
Jörg 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

View reading list on Talis Aspire

Subject specific skills

This module teaches students to carry out fundamental calculations with matrices, including the theory and computation of the Jordan canonical form of matrices and linear maps; bilinear forms, diagonalizing quadratic forms, and choosing canonical bases for these. After that the module introduces the notion of tensor, treating them rigorously.

Transferable skills

The algorithmic techniques taught have widespread "real world" applications. Examples include ranking in search engines, linear programming and optimisation, signal analysis, and graphics. To also include: clear and precise thinking; the ability to follow complex reasoning; constructing logical arguments, and exposing illogical ones; and formulating problems as algorithms, thereby enhancing understanding of details and rendering them suitable for computer implementation.