MAGIC013: Matrix Analysis

Course details

Semester

Spring 2008
Monday, January 21st to Friday, March 14th; Monday, April 28th to Friday, May 16th

Hours

Live lecture hours
20
Recorded lecture hours
0
Total advised study hours
0

Timetable

Tuesdays
14:05 - 14:55
Tuesdays
15:05 - 15:55

Description

This is offered as a core course for Applied.
Matrix theory is an active research field. It is also an important component in many areas of applied mathematics - numerical analysis, optimisation, statistics, applied probability, image processing, ...

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)

Lecturer

Bibliography

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Assessment

Attention needed

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Files

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Lectures

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