I was trying to obtain the posterior distribution of two multivariate normal distributions, but I don't know what operations are performed to go from one step to another
$X=(X_1, X_2, ... , X_n) ∼ N(\theta,\sum) $ and $ \theta∼N(\mu,A) $ (Here $\theta$ and $\mu$ are n-vectors, while $\sum$ and $A$ are nxn positive definite matrices)
Doing the conjugation and doing the algebra I get to this part
=exp{$ -\frac{1}{2}[\theta^T(n\sum^{-1} + A^{-1})\theta-2\theta^T(n\sum^{-1} \bar{x}+A^{-1}\mu)] $]
but I'm supposed to get to the next expression
=exp{$ -\frac{1}{2}(\theta-\mu_n)^T\sum_n^{-1}(\theta - \mu_n) $}
where:
$\mu_n = (A^{-1}+n\sum^{-1})^{-1}(A^{-1}\mu +n\sum^{-1}\bar{x}) $
and
$\sum_n^{-1} = (A^{-1} +n\sum^{-1})^{-1} $
but I tried everything and I can't, help!!!!!!!