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.