What is the volume of these special n-dimensional ellipsoids?

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I am in need to know the volume of a n-dimensional ellipsoid, but the matrix from which the principal axis are calculated is a singular system. This might sound strange, nevertheless, the explanation in the linked pdf will show you what I am talking about.

I guess that the solution is pretty easy for perople with a background from geometry, algebra, or vector-spaces, but I lack of a background in the respective fields of mathematics.

Since the question is pretty long I found it more convenient for you to formulate it in a http://www.energyefficiency.at/dokumente/upload/ellipsoid-volume-question_9ea7d.pdf. Nevertheless, if forum members wish that I write it in here I will do of course.

Any ideas are highly appreciated!

Kind Regards, Johannes
PS: I asked this question in another forum and offered a dinner to the person who helps me with the solution, but I was almost immediatly thrown out since people found I would ruin the spirit of the forum with that behaviour (so I won't make that same failure here again ;-) ). Thus, just to be clear: the solution to the above problem is the last unsolved part in a larger private reaseach effort in applied statistics. So its not a homework or contract work or anything alike.
 
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