I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse.
Take the Gaussian unitary ensemble (GUE) of $n \times n$ Hermitian matrices. A matrix $H$ from the GUE has diagonal entries that are real and independent $N(0,1)$, while off-diagonals are of form $X + iY$ with $X,Y$ independent $N(0,\frac{1}{2})$. All entries are independent subject to the matrix being Hermitian.
I have also seen the density of a $H$ described in terms of the Lebesgue measure on $\mathbb{R}^2$, i.e. the joint measure on the space of its $n^2$ entries: $$ P(H) \;\propto\; \exp \Big(-\frac{n}{2} \mathrm{Tr}\, H^2 \Big) $$
I can kind of sense why this is just by the fact that the matrices are Hermitian and the entries are all independent (subject to the matrix being Hermitian) so you can express the joint as a product, but I'm not sure how one rigorously shows this...
Thanks so much,