I found this method somewhere (I don't remember where) and I'm wondering whether it is correct. One has an ellipsoid of any dimension and one wants to uniformly generate some points at random on its boundary.
Uniformly simulate some points on the sphere (with same dimension as the ellipsoid) $x_1$, $x_2$, $\ldots$
Take the upper Cholesky factor $U$ of the shape matrix $A$ of the ellipsoid ($x'Ax = r^2$)
Then the solution $Y$ of $UY = (x_1 \, x_2 \, \ldots)$ is a matrix of uniformly random points on the surface of the ellipsoid.
I checked in dimension 2: I performed this simulation and I calculated the proportion of points falling in a given arc, then I compared with the arc length. The two results were close but not very close, so I suspect this way is wrong.
Now, what about simulating in the volume bounded by the ellipsoid by adapting the previous method:
Uniformly simulate some points in the sphere (with same dimension as the ellipsoid) $x_1$, $x_2$, $\ldots$
Take the upper Cholesky factor $U$ of the shape matrix $A$ of the ellipsoid ($x'Ax \leqslant r^2$)
Then the solution $Y$ of $UY = (x_1 \, x_2 \, \ldots)$ is a matrix of uniformly random points in the ellipsoid.
Is it correct? Maybe this is only one of these two methods that I found somewhere, and perhaps one is correct.
