MAGIC013: Matrix Analysis

Course details


Autumn 2009
Monday, October 5th to Friday, December 11th


Live lecture hours
Recorded lecture hours
Total advised study hours


11:05 - 11:55
11:05 - 11:55


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


No prerequisites information is available yet.


  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)



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