This is a very simple question but I can't find the derivation anywhere on the internet or in a book. I would like to see the derivation of how one Bayesian updates a multivariate normal distribution. For example: imagine that
$$ P({\bf x}|{\bf μ},{\bf Σ}) = N({\bf \mu}, {\bf \Sigma}) \\ P({\bf \mu}) = N({\bf \mu_0}, {\bf \Sigma_0}) $$
After observing a set of ${\bf x_1 ... x_n}$, I would like to compute $P({\bf \mu | x_1 ... x_n})$. I know that the answer is $P({\bf \mu | x_1 ... x_n}) = N({\bf \mu_n}, {\bf \Sigma_n})$ where
$$ \bf \mu_n = \Sigma_0 (\Sigma_0 + \frac{1}{n}\Sigma)^{-1}(\frac{1}{n}\sum_{i=1}^{n}{\bf x_i}) + \frac{1}{n}\Sigma(\Sigma_0+\frac{1}{n}\Sigma)^{-1}\mu_0 \\ \bf \Sigma_n = \Sigma_0(\Sigma_0 + \frac{1}{n}\Sigma)^{-1}\frac{1}{n}\Sigma $$
I am looking for the derivation of this result with all the intermediate matrix algebra.
Any help is much appreciated.
