I am stuck on this problem:
Let $X$ be a $\mathbb{R}^n$-valued random variable with zero mean and covariance matrix $\Sigma$. Show that
$h(X) \le \frac{1}{2}log(2 \pi e)^{n} |\Sigma|$
with equality iff $X$ is multivariate normal.
I believe that, treating the variable as $n$ $\mathbb{R}$-valued random variables,
$h(X) \le \sum\limits_{i=1}^n h(X_i)$, with equality iff the $X_i$'s are pairwise independent.
However I need some help understanding the meaning of the problem, and where the normal distribution is introduced.
Thanks for your help.