# PhD Course 2009

## An Introduction to Pseudospectra and their Applications

### Lecture 3

The program for the third day is as follows:

12:30-13:00
Questions and answers. Review of results on pseudospectra obtained last time. Review slides are here.
13:05-14:00
Further results on pseudospectra. I will cover pages 19-20 and example 6.4, pages 24-26, from the notes, in detail. Then I will explain how to define functions of a matrix. This covers pages 14 and 15 in the notes.
14:00-14:20
Coffee break.
14:20-15:10
Exercises, see below.
15:10-15:15
Break.
15:15-16:00
Lecture on the properties of norm(exp(tA)). Relation to spectrum, numerical range and pseudospectra. I will cover the material on pages 34-40 in the notes, although I will not talk about the proofs in detail. I hope to have time for some examples also.

#### Exercises

Theoretical exercises

From the notes the following exercises:

• Exercise 2.4, mainly for mathematics and physics graduate students
• Exercise 2.8, mainly for mathematics and physics graduate students
• Exercise 4.2, an easy calculation
• Exercise 4.4, almost as easy as the previous calculation
• Exercise 4.16, recommended to all participants
• Exercise 4.17, also recommended to all participants
• Exercise 6.3, must be done by everyone.

Eigtool exercises

If you have not yet tried the various possibilities in eigtool (both the built-in demos and pseudospectra of matrices you define), now is the time to do so. From tomorrow I assume that you are familiar with the basics of this toolbox. Choose various matrices and experiment with them. Choose both normal and non-normal matrices as examples. I can provide suggestions for this part.

Further exercises using eigtool:

• Exercise 6.4, must be done by everyone
• Exercise 6.6, this is a more challenging exercise. Remember that you can get the pseudospectra in a separate figure (choose from the file menu 'printable plot'), and then plot circles on top of the pseudospectra, by using the 'hold on' command.

Webpages maintained by Arne Jensen, matarne at math.aau.dk
Updated 15/09/2009, 12:55