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.
At level 7, topics will be studied and assessed to greater depth.
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).
Additional reading for level 7:
G. Teschl: Topics in Real and Functional Analysis. Lecture notes available at http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of the theory of linear operators;
2 demonstrate the capability to solve complex problems using a very good 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 a range of concepts and principles in Hilbert space theory and operator theory in loosely defined contexts, showing good judgment in the selection and application
of tools and techniques.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
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 effectively;
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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