Transcript of 15_02.AVI (Lecture 15, part 2)

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

... Prof. Nagaoka ... Yes, please have a look at the computer graphics. And here you see the Jordan normal form of A in its decomposition as a diagonal matrix plus a nilpotent matrix. And now we take it times x, this scalar x, and because x is just a scalar, we can also commute it with the inverse of P and write it here as x A. And this becomes on the right side x times the diagonal matrix D, plus x times the nilpotent matrix N. And we can lift this into the exponential form. And then on the right side we have the exponential of x D plus x N. And we can again split this into two factors, exponential x D times exponential x N, because the two matrices D and N commute.

... Prof. Nagaoka ...

And as before we can use the fact that N is nilpotent, the square of N is zero, and therefore the exponential x N just has the two terms, the unit matrix E plus x times N.

... Prof. Nagaoka ...

And x D is a diagonal matrix and as you already know, the exponential of the diagonal matrix can be expressed with the terms only in the diagonal entries. And this is the matrix E plus x N on the right. If we calculate the product, we just get a new term in this off diagonal position, x times e to the power of two x. And now as before, we can pull the inverse of P and P to the right and left out of the exponential and we multiply with P from the left and with the inverse of P from the right. And if we explicitly insert P and the inverse of P, and conduct the calculation, we get a matrix, which as its entries always has linear combinations of:
e to the power of two x,
x times e to the power of two x,
and e to the power of three x.

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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.