Say I have an improper multivariate normal distribution with singular precision matrix $Q$. I can compute the pseudo-inverse of $Q$ and use this as a covariance matrix, i.e. $\Sigma = Q ^{+}$. I seem to be able to sample from this distribution using the standard approach of transforming a sample from a standard normal using some decomposition of $\Sigma$.
But what am I actually sampling from here? Since this is improper there must be some effective constraint on the kernel of $Q$. What is that constraint? Is it any consequence of taking a pseudo-inverse or is it just the usual limits of what my computer can actually sample from as an approximation of something improper?
More generally, how can I think about using a pseudo-inverse parameterisation beyond just thinking to myself, "the inverse doesn't exist so I use the pseudo-inverse instead"? This is basically what I do when I use a pseudo-inverse.
