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General


Description

This is offered as a core course for Applied.
 
Matrix theory is an active research field, with at least international four journals devoted to the subject. It is also an important component in many areas of applied mathematics - numerical analysis, optimisation, statistics, applied probability, image processing, ...
The solution to many problems in Science, Engineering and Mathematics lies in a "matrix fact". Dennis Bernstein

Semester

Autumn 2009 (Monday, October 5 to Friday, December 11)

Timetable

  • Mon 11:05 - 11:55
  • Fri 11:05 - 11:55

Prerequisites

No prerequisites information is available yet.

Syllabus

  1. Introduction (2 lectures)
    • Matrix products - Standard product, tensor/Kronecker product, Schur product
    • Decompostions - Schur form, Real Schur form, Jordan form, Singular Value decompositions
    • Other preliminaries - Schur complement, additive and multiplicative compounds
  2. Norms (3 lectures)
    • norms on vector spaces
    • inequalities relating norms
    • matrix norms
    • unitarily invariant norms
    • numerical radius
    • perturbation theory for linear systems
  3. Gerschgorin's Thorem, Non-negative matrices and Perron-Frobenius (4 lectures)
    • diagonal dominance and Gerschgorin's Theorem
    • spectrum of stochastic and doubly stochastic matrices
    • Sinkhorn balancing
    • Perron-Frobenius Theorem
    • Matrices realted to non-negative matrices - M-matrix, P-matrix, totally positive matrices.
  4. Spectral Theory for Hermitian matrices (2 lectures)
    • Orthogonal diagonalisation
    • Interlacing and Monotonicity of Eigenvalues
    • Weyl's and the Lidskii-Weilandt inequalities
  5. Singular values and best approximation problems (2 lectures)
    • Connection with Hermitian eigenvalue problem
    • Lidskii-Weilandt - additive and multiplicative versions
    • best rank-k approximations
    • polar factorisation, closest unitary matrix, closest rectangular matrix with orthonormal columns
  6. Positive definite matrices (3 lectures)
    • Characterisations
    • Schur Product theorem
    • Determinantal inequalitties
    • semidefinite completions
    • The Loewner theory
  7. Perturbation Theory for Eigenvalues and Eigenvectors (2 lectures)
    • primarily the non-Hermitian case
  8. Functions of matrices (2 lectures)
    • equivalance of definitions of f(A)
    • approximation of/algorithms for general functions
    • special methods for particular functions (squareroot, exponential, logarithm, trig. functions)

Students


Photo of Hanefa Al-Qasmi
Hanefa Al-Qasmi
(Manchester)
Photo of Fatih Bribesh
Fatih Bribesh
(Loughborough)
Photo of Lini Cao
Lini Cao
(York)
Photo of Tim Crinion
Tim Crinion
(Manchester)
Photo of Aiman Elragig
Aiman Elragig
(Exeter)
Photo of Betty Fyn-Sydney
Betty Fyn-Sydney
(Birmingham)
Photo of James Hook
James Hook
(Manchester)
Photo of Raja Zafar Iqbal
Raja Zafar Iqbal
(Birmingham)
Photo of Paul  Jones
Paul Jones
(Loughborough)
Photo of Avais Kasim Sait
Avais Kasim Sait
(Lancaster)
Photo of Sajida Kousar
Sajida Kousar
(York)
Photo of Antony Lee
Antony Lee
(Nottingham)
Photo of Qifeng Liao
Qifeng Liao
(Manchester)
Photo of Warren Lockhart
Warren Lockhart
(Liverpool)
Photo of Parsons Mark
Parsons Mark
(Reading)
Photo of Rudaben Meskarian
Rudaben Meskarian
(Southampton)
Photo of Samantha Newsham
Samantha Newsham
(Lancaster)
Photo of Mobolaji Osinuga
Mobolaji Osinuga
(Manchester)
Photo of Merlijn van Horssen
Merlijn van Horssen
(Nottingham)
Photo of Chris Welshman
Chris Welshman
(Manchester)


Bibliography


No bibliography has been specified for this course.

Assessment



No assessment information is available yet.

No assignments have been set for this course.

Files


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Week(s)File
Matrix-Analysis-Notes-complete.pdf


Recorded Lectures


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