# Non-informative prior for the covariance matrix

I'm currently working on a project around the Bayesian approach to portfolio selection, and I can't manage to wrap my mind around the specification of the non-informative (diffuse) prior. Assuming gaussianity of the returns with parameters $$\mu \in R^N$$ and $$\Sigma \in R^{N x N}$$ the prior is usually expressed as follows:

$$p(\mu,\Sigma) \propto |\Sigma|^{-(N+1)/2}$$

Where $$|\Sigma|$$ is $$det(\Sigma)$$ and $$N$$ the number of variables (assets in my case). I understand that this comes from the marginal priors of $$\mu$$ and $$\Sigma$$, given by

$$p(\mu) \propto c$$

$$p(\Sigma) \propto |\Sigma|^{-(N+1)/2}$$

I understand the specification of $$p(\mu)$$, so that's not a problem. However, I do not understand the specification of $$p(\Sigma)$$. I understand that in the special case $$N=1$$, we obtain the usual prior for scale parameters $$p(\sigma) \propto 1/\sigma$$, but I don't get how we obtain $$|\Sigma|^{-(N+1)/2}$$.

I've checked already well-known references: Zellner, 1971 / Bawa, Brown and Klein 1974 / Geisser 1965 / etc but it doesn't help. It seems that there is something that I fundamentally don't grasp.

If anyone has an intuitive and clear explanation for this, it would be great !!

Many thanks in advance.