Authorised by Academic Registrar, April 1996
Objectives On the completion of this subject students will understand some of the mathematical structure possessed by matrices; appreciate how abstract notions like vector spaces and dimension are powerful tools for studying objects like matrices; have improved skill at computations with eigenvalues and eigenvectors; know of the mathematical technique of expressing general objects (like matrices) in terms of elementary objects (like diagonal matrices, or triangular matrices, or idempotent matrices) and see why this is useful; understand Schur's unitary triangularisation theorem and some of its many consequences; learn about norms, inner products and metrics on spaces of matrices; see how the elementary tools of analysis can be used to locate eigenvalues, to analyse computational errors, and to provide least squares approximate solutions to systems of equations.
Synopsis Linear spaces. Eigenvalues and eigenvectors. Schur's unitary triangularisation theorem and its consequences. Spectral theorems. Norms on vectors and matrices. Functions of matrices. Application to error analysis. Least squares solutions to linear equations. Location and perturbation of eigenvalues.
Assessment Examinations (1.5 hours): 80% + Assignments: 20%