Aalborg University
The International Doctoral School of Technology and Science

PhD Course 2009

An Introduction to Pseudospectra and their Applications

Course title
An Introduction to Pseudospectra and their Applications
Professor Arne Jensen, Department of Mathematical Sciences, Aalborg University. matarne at math.aau.dk
  1. September 10, 2009, 12:30-16:00
  2. September 11, 2009, 8:30-12:00
  3. September 17, 2009, 12:30-16:00
  4. September 18, 2009, 8:30-12:00
  5. September 25, 2009, 8:30-12:00
  6. September 28, 2009, 12:30-16:00
  7. October 5, 2009, 12:30-16:00
The lectures take place at the Department of Mathematical Sciences, Fredrik Bajers Vej 7G, Room G5-109

Short course description

The concept of pseudospectra of matrices and operators is relatively recent, but has gained a wide acceptance as a useful method to understand properties of non-normal (in particular non-Hermitean) matrices. For example, a linear system governed by a non-normal matrix may exhibit a behavior at intermediate times which is very different from the initial behavior and the large time asymptotic behavior. The pseudospectra allows one to quantify this behavior. Another application is the analysis of the stability of numerical differentiation methods, used to discretize differential equations.

The participants can choose to concentrate on the theory or the applications. They are expected to analyze a number of systems based on either matrices from their own projects, or provided by the lecturer. A Matlab toolbox for computing and analyzing pseudospectra is available, see below.

Prerequisites are a good knowledge of linear algebra, preferably including numerical linear algebra, and some familiarity with Matlab.

The course will be based on lecture notes by the lecturer. A preliminary version is available (version of August 23, 2009):

The material on polynomial interpolation and spectral differentiation can be found here. Note that on this web site you can find some chapters from the book

N. L Trefethen, Spectral Methods in Matlab. SIAM 2000

and a link to the important paper

J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange Interpolation, SIAM Review 46 (2004), 501-517.

Furthermore, a number of m-files can be downloaded from the site.


An extensive presentation of the theory and the applications of psseudospectra can be found in the book

N. L. Trefethen and M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators. Princeton University Press 2005.

A mathematically oriented introduction to pseudospectra can be found in the following book. The book also gives an introduction to much of the operator theory and spectral theory needed for a mathematically rigorous treatment of pseudospectra.

E. B. Davies, Linear Operators and their Spectra. Cambridge University Press 2007.


The pseudospectra of matrices can be computed numerically. A free toolbox called EigTool for Matlab can be found here. We will use this toolbox extensively during the course, so it is recommended to download and install it on your computer.

On the lectures

Information on the lectures will be given here.

Date Time Topics
September 10 12:30-16:00 Introduction to the course. A quick slide overview of the course. Some results from linear algebra and operator theory. Schedule.
September 11 08:30-12:00 Definition of pseudospectra. Basic properties of pseudospectra. An introduction to eigtool. Exercises. Schedule.
September 17 12:30-16:00 Questions and answers. Review of results on pseudospectra from last week. Further results on pseudospectra. Transient behavior of exp(tA). Exercises. Schedule.
September 18 08:30-12:00 Questions and answers. Review of results on norm(exp(tA)). Examples. Exercises concerning norm(exp(tA)). Results on norm(An) for n large. Schedule.
September 25 08:30-12:00 Questions and answers. Brief introduction to polynomial interpolation. Matrices arising from numerical differentiation. Examples. Exercises concerning the theory and also Chebyshev differentiation matrices. Schedule.
September 28 12:30-16:00 Questions and answers. About presentations October 5. Perturbation theory and pseudospectra. Examples. Exervises. Schedule.
October 5 12:30-16:00 Presentations by the participants. Pointers to further reading. Schedule.

The links under the heading Topic will be established, as the course progresses.


Each lecture will comprise a number of exercises. In order to carry out these exercises you should bring with you a laptop with MATLAB and Eigtool installed. Detail of the exercises will be given in connection with each lecture.


A number of files will be put on the web. These are intended as starting points for your own experiments. They will also be used in the exercises.

Evaluation of the course

In order to obtain credit for the course two things are required. One is active participation in the lectures and exercises. The other is a short presentation to be given on the last day of the course.

This presentation can be practical, i.e. application of the results on pseudospectra to a problem that is part of your PhD project, or it can be theoretical, dealing with aspects of pseudospectra not covered in the course.

The presentations will be discussed, as the course progresses. I will be glad to suggest topics for presentation.

Webpages maintained by Arne Jensen, matarne at math.aau.dk
Updated 30/09/2009, 16:55