Given a random variable $X \in \mathbb{R}^n$, with variance/covariance matrix $\Sigma$ and mean 0, how do we prove the following identity? $$ \mathbb{E}(X^T\Sigma^{-1}X) = n = \dim X$$
Does this identity have a name? It was provided as a hint, to be used without proof, to prove that the multivariate normal distribution has maximum differential entropy for a given variance matrix $\Sigma$.
$$ \mathbb{E}(X^T\Sigma^{-1}X) = \mathbb{E}(\mathrm{Tr} (X^T\Sigma^{-1}X))=\mathbb{E}(\mathrm{Tr} (XX^T\Sigma^{-1}))=\mathrm{Tr}(\mathbb{E}(X X^T)\Sigma^{-1}) = \mathrm{Tr}(\Sigma \Sigma^{-1}) = n$$