MAS8712 : Matrix Analysis & Representation Theory (Inactive)
- Inactive for Year: 2024/25
- Module Leader(s): Dr David Kimsey
- Lecturer: Dr Martina Balagovic
- Owning School: Mathematics, Statistics and Physics
- Teaching Location: Newcastle City Campus
Semesters
Your programme is made up of credits, the total differs on programme to programme.
Semester 1 Credit Value: | 10 |
Semester 2 Credit Value: | 10 |
ECTS Credits: | 10.0 |
European Credit Transfer System |
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 Methods
Teaching Activities
Category | Activity | Number | Length | Student Hours | Comment |
---|---|---|---|---|---|
Structured Guided Learning | Lecture materials | 36 | 1:00 | 36:00 | Non-Synchronous Activities |
Scheduled Learning And Teaching Activities | Lecture | 9 | 1:00 | 9:00 | Synchronous On-Line Material |
Guided Independent Study | Assessment preparation and completion | 30 | 1:00 | 30:00 | N/A |
Scheduled Learning And Teaching Activities | Lecture | 9 | 1:00 | 9:00 | Present in Person |
Structured Guided Learning | Structured non-synchronous discussion | 18 | 1:00 | 18:00 | N/A |
Scheduled Learning And Teaching Activities | Drop-in/surgery | 4 | 1:00 | 4:00 | Office Hour or Discussion Board Activity |
Guided Independent Study | Independent study | 94 | 1:00 | 94:00 | N/A |
Total | 200:00 |
Jointly Taught With
Code | Title |
---|---|
MAS3712 | Matrix Analysis & Representation Theory |
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 Methods
The format of resits will be determined by the Board of Examiners
Exams
Description | Length | Semester | When Set | Percentage | Comment |
---|---|---|---|---|---|
Written Examination | 120 | 2 | A | 80 | Alternative assessment - class test |
Other Assessment
Description | Semester | When Set | Percentage | Comment |
---|---|---|---|---|
Written exercise | 1 | M | 8 | written exercises |
Written exercise | 2 | M | 12 | written exercises |
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.
Reading Lists
Timetable
- Timetable Website: www.ncl.ac.uk/timetable/
- MAS8712's Timetable