# MATRIX ANALYSIS - 2020/1

Module code: MAT3045

## Module Overview

This module continues to develop the theory of matrices initiated in MAT1034 Linear Algebra, and introduces concepts of convergence and distance to the study of matrices. The module then progresses through several advanced topics in linear algebra such as the Perron-Frobenius theorem, tensors and exterior powers of matrices, and singular value decompositions. As an application we will investigate the PageRank algorithm used by Google to rank web pages.

### Module provider

Mathematics

MORRIS Ian (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 117

Lecture Hours: 33

Semester 1

## Prerequisites / Co-requisites

MAT1034 Linear Algebra, MAT2004 Real Analysis 2.

## Module content

Indicative content includes:

• The Jordan normal form

• Norms; equivalence of norms in finite dimensions; convergence of sequences and series of matrices; operator norms

• The singular value decomposition

• The spectral radius formula and Yamamoto's theorem

• Positivity and the Perron-Frobenius theorem

• The PageRank algorithm

• Tensors and exterior powers of matrices

## Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION 80
School-timetabled exam/test CLASS TEST (50 MINS) 20

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and manipulate mathematical statements.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One two hour examination at the end of Semester 2; worth 80% module mark.

• One 50 minute class test; worth 20% module mark

Formative assessment and feedback

Students receive written feedback via a number of marked coursework assignments over an 11 week period. In addition, verbal feedback is provided by lecturer at biweekly tutorial lectures.

## Module aims

• Extend and deepen students' understanding of linear maps on finite-dimensional spaces
• Familiarise students with important matrix decompositions, such as the Jordan form and singular value decomposition
• Familiarise students with core topics in the analysis of matrices such as convergence, norms and the Perron-Frobenius theorem
• Introduce students to tensor and exterior algebra of matrices

## Learning outcomes

 Attributes Developed 001 Rigorously interpret convergence and infinite summation of matrices, including matrix exponentials KC 002 Define and calculate singular values, Kronecker products and exterior powers, at least in simple cases KC 003 Understand the interaction between the norm/singular values of a matrix or matrices and the tensor / exterior product KC 004 Understand the significance of positivity and the Perron-Frobenius theorem. 005 Define the determinant using exterior algebra and derive its essential properties KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• An introduction to analytic methods in linear algebra and their applications.

The learning and teaching methods include:

• 3 x 1 hour lectures per week x 11 weeks, with additional notes on white board to supplement the module handbook and Q + A opportunities for students.

(every second week) 1 x 1 hour tutorial replaces one of the lectures for guided discussion of solutions to problem sheets provided to and worked on by students during the tutorial.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.