Module Catalogue 2025/26

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

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.

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.

Intended Knowledge Outcomes

Students will learn various techniques for performing algebraic and analytic manipulations on matrices. By the end of the course they will be able to understand the basic terminology of matrix analysis, diagonalise matrices, classify matrices with respect to the Jordan form and solve eigenvalue problems. Students will see applications of these methods to real-life problems, for example, solving systems of ordinary differential equations, image compression and Google’s PageRank algorithm.

Intended Skill Outcomes

Students will enhance their ability to understand how eigenvalue problems may arise in practical problems. They will learn to recognise which technique is appropriate each time to provide solutions and efficient approximations.

Students will develop skills across the cognitive domain (Bloom’s taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments 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.

General Notes

N/A