Transcript of 15_01.AVI (Lecture 15, part 1)

By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

Yes, please have a look at the computer graphics. Here you see the matrix A, which you saw in the differential equation. And next is the matrix P, which serves for transforming A into its Jordan normal form: inverse P A P. And the Jordan normal form can be decomposed into a diagonal matrix and a nilpotent matrix: D and N. And if you calculate the square of N it becomes zero, because there was just one in the off diagonal position.

Now the exponential of the Jordan normal form is just the exponential of D plus N. And this can be factorized into exponential of D, times exponential of N, because the two matrices D and N commute.

... Prof. Nagaoka ...

Yes, and we can expand the exponential of N into the sum over the N to the power of zero, one, two, and so one. And then we use the fact that N is nilpotent, and so this sum actually will be cut off after the first two terms. So writing the first two terms explicitly, it is the zeroth power, which is the unit matrix, and the first power, which is N itself. And now we can insert the diagonal matrix, and E plus N. And if we calculate the product, we just get another e squared in this off diagonal position.

... Prof. Nagaoka ...

And now we take the inverse of P and P out of the exponential, and we multiply with P from the left and with the inverse of P from the right. And we insert P and the inverse of P. And then we calculate the resulting matrix. And it is a matrix, which has as entries simply linear combinations of e squared, and e to the power of three.

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Soli Deo Gloria. Created by Eckhard Hitzer (Fukui).
Last Modified 16 January 2004
EMS Hitzer and Prof. R. Nagaoka are not responsible for the content of external internet sites.