This module is not currently running in 2024 to 2025.
Matrix theory: Hermitian and symmetric matrices, spaces of these matrices and the associated inner product, diagonalization, orthonormal basis of eigenvectors, spectral properties, positive definite matrices and their roots
Hilbert space theory: inner product spaces and Hilbert spaces, L^2 and l^2 spaces, orthogonality, bases, Gram-Schmidt procedure, dual space, Riesz representation theorem
Linear operators: the space of bounded linear operators with the operator norm, inverse and adjoint operators, Hermitian operators, infinite matrices, spectrum, compact operators, Hilbert-Schmidt operators, the spectral theorem for compact Hermitian operators.
Additional topics, may include:
- the Rayleigh quotient and variational characterisations of eigenvalues,
- the functional calculus,
- applications to Sturm-Liouville systems.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
Level 6
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Level 7
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Reassessment methods:
Like-for-like
J.R. Giles: Introduction to the Analysis of Normed Linear Spaces. Cambridge University Press (2000).
V.L. Hansen: Functional Analysis – Entering Hilbert Space. World Scientific (2006).
R. Horn , C. Johnson: Matrix Analysis. Cambridge University Press (1985).
C.D. Meyer: Matrix Analysis and Applied Linear Algebra. SIAM (2000).
B. Rynne, M. Youngson: Linear Functional Analysis. Springer (2008).
G. Strang: Linear Algebra and its Applications, 3rd edition. Saunders (1988).
N. Young: An Introduction to Hilbert space. Cambridge University Press (1988).
F. Zhang: Matrix Theory – Basic Results and Techniques. Springer (2011).
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of matrix and operator theory;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of
the material in the following areas: Hermitian matrices and their spectral properties, Hilbert spaces, linear operators and functionals, compact operators, spectral theory;
3 apply key aspects of operator theory in well-defined contexts, showing judgement in the selection and application of tools and techniques.
The intended generic learning outcomes. On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 communicate technical material competently;
7 demonstrate an increased level of skill in numeracy and computation;
8 demonstrate the acquisition of the study skills needed for continuing professional development.
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