I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this (here) a user suggested that I compute the Cholesky decomposition of the matrix and then:
sample Gaussian with identity matrix as the covariance (easy to do by sampling each coordinate as a Gaussian) and then apply the linear transformation from the Cholesky decomposition matrix to get your sampled point from the ellipsoid covariance Gaussian
I've had a go at following this but I assume I've misunderstood to at least some extent as the result isn't consistent with what I'd expect. The steps I took were:
1) Calculate the Cholesky decomposition of the covariance matrix.
2) Sample each initial vertex point as a Gaussian with width 1 to generate (x', y', z')
3) Multiply (x',y',z') by the Cholesky decomposition matrix for the newly generated point.
I'm certain this isn't correct, but don't have the experience to know exactly what is wrong.
Thanks in advance!