Maximum-likelihood estimate for multivariable normal distribution

I have to estimate the empirical mean value, covariance and correlation given a sample set $X_i$, $i=1, \dotsc , N$. I am trying to derive the mean value and I arrived at this point: $$\sum_{i=1}^N {\boldsymbol \Sigma}^{-1} (x_i - \mu) = 0.$$ How can i continue?

• I think this might qualify for a self-study tag since you seem to want some hints as to how to proceed, but not the final answer? Anyway, my suggestion would be to try distributing the multiplication with $\Sigma^{-1}$ over the $x_i$ and $\mu$, and then see if you can get it from there. – Ruben van Bergen Dec 14 '16 at 11:52
• @RubenvanBergen Thanks. Yes, just some hints. Do I have to know any particular property of the covariance matrix in order to continue? Thanks. – wrong_path Dec 14 '16 at 11:58
• Nope. But let me correct my previous comment as I just realized that wasn't really helpful as a first step. On second thought the first thing you really need to know at this point to proceed is that you can take the multiplication by $\Sigma^{-1}$ out of the sum over $i$. And then (without wanting to give away too much) you need to think about how to get rid of it ;). – Ruben van Bergen Dec 14 '16 at 13:38
• @RubenvanBergen Thanks a lot, I was able to derive an estimate for both the mean value and the covariance. One last thing: the problem is now with the correlation: is it possible to estimate it with a similar process? Or it derives from the estimate for the covariance? Again, thank you. – wrong_path Dec 14 '16 at 17:43