MA5001 Mathematics  Advanced Linear Algebra
Semester 1 201718
This module is for students in the coursebased M.A. and M.Sc. in Mathematics at NUI Galway, and for research students in the School of Mathematics, Statistics and Applied Mathematics. All interested students are very welcome to participate. Lectures begin on September 4 2017. The Course Outline document is here.
Lecturers: 
Dr Rachel Quinlan, Room
ADBG007, Arás de Brún.
Email: rachel.quinlan@nuigalway.ie


Dr Niall Madden, Room
ADB1013, Arás de Brún.
Email: niall.madden@nuigalway.ie

Lectures: 
Monday 11:0011:50 in ADB1020
Tuesday 12:0012:50 in ADB1020
Wednesday 12:0012:50 in ADB1020 
Course Content 
This course is an exploration of some topics in linear algebra, from algebraic and algorithmic viewpoints. Linear algebra is a vast subject that pervades virtually all areas of mathematical work including (for example) algebra, analysis, geometry, coding theory, differential equations, numerical analysis, computational mathematics and mathematical modelling.
Mostly, we will investigate the spectrum (or list of eigenvalues) of
a square matrix. We will consider such questions as why we should care
about the spectrum, how to calculate a spectrum, and what can be
deduced about the spectrum from structural properties of a matrix or
from inspection of its entries. Both theoretical and computational
considerations will be included, and we will discuss connections to
graph theory and to other areas of mathematics and applied
mathematics.
The course will include some classical highlights of matrix theory and some of their implications, for example
 The Cauchy Interlacing Theorem
 The PerronFrobenius Theorem
 The Gershgorin Circle Theorem(s)
 The Jordan Canonical Form
 The Singular Value Decomposition.
 The Schur form
 The Power method
 The QR factorisation algorithm.

Assessment: 
There will be three or four homework assignments during the semester, as well as a single 2hour exam in the Winter exam session.

Outline lecture notes will be posted here in instalments as the module proceeds.

Lecture 1: Linear Transformations and Eigenvalues(RQ, 4 September)

Lecture 2: Multiplying a matrix by a vector
(NM, 5 September). This closely followed
Trefethen and Bau: Lecture 1.

Lecture 3: Range, nullspace, and rank
(NM, 6 September). More from
Trefethen and Bau: Lecture 1.

Lecture 4: The matrix of a linear transformation
(RQ, 11 September)

Lecture 5: Some bases are better than others
(RQ, 12 September)

Lecture 6: Orthogonal vectors
(NM, 12 September).

Lecture 7:
Unitary matrices
(NM, 18 September)

Lecture 8 and 9:
Real symmetric matrices
(RQ, 19 and 20 September)

Lecture 10:
Vector norms
(NM, 25 September)

Lecture 11:
Matrix norms
(NM, 26 September)

Lecture 12:
Diagonalizability of real symmetrix matrices
(RQ, 27 September)

Lecture 13:
Introduction to the Singular Value Decomposition
(NM, 2 October)

Lecture 14:
No class, because of the workshop on tutoring in Mathematics and Statistics at NUI Galway.

Lecture 15:
Tutorial.
(RQ, 4 October)

Lecture 16 and 17:
Positive (semi)definite matrices
(RQ, 9 and 10 October)

Lecture 18:
Existence of the SVD
(NM, 11 October)

Lecture 19:
No class
.
(Ophelia, 16 October)

Lecture 20:
Yet more about the SVD.
(NM, 17 October)

Lecture 21:
(RQ, 18 October)

Lecture 22:
Some theorems about the SVD.
(NM, 23 October)

Lecture 23:
Low rank approximations.
(NM, 24 October)

Lecture 24:
Positive matrices  the PerronFrobenius Theorem.
(RQ, 25 October)

Lecture 25:
Proof of the PerronFrobenius Theorem (I)
(RQ, 31 October)

Lecture 26:
Projectors (I).
(NM, 1 November)

Lecture 27:
Orthogonal Projector.
(NM, 6 November)

Lectures 28 and 29:
Proof of the PerronFrobenius Theorem (II + III)
(RQ, 7 and 8 November)

Lecture 30:
QR Factorisation.
(NM, 13 November)

Lecture 31:
The Schur Form.
(NM, 14 November)

Lectures 32 and 33:
Applications of PerronFrobenius theory (I + II)
(RQ, 15 and 20 November)

Lecture 34:
Hessenberg Form and the QR Algorithm.
(NM, 21 November)

Lecture 35:
Review.
(NM + RQ, 22 November)
The notes from the versions of this module that were taught by RQ and ran in 15/16 and 16/17 are
here and
here (but the content changes from year to year).