# Aalborg University

The International Doctoral School of Technology and Science

# PhD Course 2011

## Mathematical Kaleidoscope Part 2

- Topic Part 2
**An Introduction to Pseudospectra and their Applications**
- Lecturer
- Professor Arne Jensen, Department of Mathematical Sciences, Aalborg University. matarne at math.aau.dk
- Date
- April 29, 2011, 12:30-16:00
- Location
- The lectures take place at the Department of Mathematical Sciences, Fredrik Bajers Vej 7G, Room G5-109

### Important:

Please bring your computer to this part of the course. Make sure you have installed Matlab. Please download the toolbox **EigTool** beforehand, see the link below.
### Short 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.

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

Lecture notes are available. These notes were also used for a course in 2009.

- Notes in .pdf format are here (app. 7.5Mb)
- Notes in zipped .ps format are here (app. 1.5Mb)

### Literature

An extensive presentation of the theory and the applications of pseudospectra 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.

### Software

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

I will start with reviewing some basic results from linear algebra and then give a quick overview of the definition and applications of pseudospectra. After that I will go into details with the properties and applications of the pseudospectra. I will concentrate on explaining the results and their interpretation. The detailed proofs of the results are available in the lecture notes.

During this session there will also be some exercises, where you will use Matlab together with EigTool to visualize pseudospectra.

### Evaluation of Part 2

Through active participation in this part of the course.

Webpages maintained by Arne Jensen, matarne at math.aau.dk

Updated 25/04/2011, 12:25