Aims

To equip students with a range of tools and methods for diagonalising and factorising matrices. To understand these techniques and applications that arise both in pure and applied sciences. To reinforce the ability of students to identify real-life problems that can be solved with matrices.

Module summary
Matrices play a key role in mathematics with many applications to pure, statistics and physics. They are necessary in almost every area of science, whether it be mathematics, economics, engineering or operational research. Matrix analysis provides a common framework to this effect. It allows the development of design tools and algorithms that solve efficiently linear systems, polynomial matrix equations, optimization problems, as well as problems that arise in quantum information theory. In this course we focus on key results that enable the combination of linear algebra with mathematical analysis. By the end of the course the students will understand classical and recent results of matrix analysis that have proved to be important to pure and applied mathematics.

The module presents the theory of complex representations of finite groups including the discussion of important classes of examples. Starting from the motivating question how a group can act linearly on a vector space, students will see an instance of a complete mathematical theory. While of major importance for the study of finite groups, this setup also forms a starting point for more general representation theory.

Outline Of Syllabus

Matrix factorisations (Jordan normal form, polar decomposition, singular value decomposition etc.).
Similarity classes of matrices.
Hermitian matrices and positive definite matrices.
Spectral theorems for normal matrices and various subclasses.
Perron-Frobenius Theorem.

Review of linear algebra (complex inner product spaces, spectral theorem)
General linear group
(Complex) representations of groups
Maschke’s Theorem
Morphisms of representation, Schur’s Lemma
Orthogonality relations
Class functions, characters, and character tables
Regular representation
Representations of abelian groups
Representations of the symmetric group

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.