The Probability Density Function of a Transformation-based   Hyperellipsoid Sampling Technique
  Sun and Farooq [2] showed that random samples can be efficiently drawn from an arbitrary n-dimensional hyperellipsoid by transforming samples drawn randomly from the unit n-ball. They stated that it was a straightforward to show that, given a uniform distribution over the n-ball, the transformation results in a uniform distribution over the hyperellipsoid, but did not present a full proof. This technical note presents such a proof. 
Jonathan D. Gammell and Timothy D. Barfoot
math.ST stat.TH