The Gaussian orthogonal ensemble of $N\times N$ symmetric matrices is often defined as a matrix whose diagonal elements are drawn from the Gaussian distribution $N(0,1)$ and off-diagonal elements from $N(0,\frac{1}{2})$.
However, in Livan's book Introduction to Random Matrices, the authors claim that this joint pdf can actually be derived from the following:
- Sample a random $N\times N$ matrix whose elements are drawn from $N(0,1)$. At this point $$p(H)=p(H_{11},\dots,H_{NN})=\prod_{i,j=1}^N\left(\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{H_{ij}^2}{2}\right]\right).$$
- Define the symmetrisation $S=\frac{1}{2}(H+H^\intercal)$, whose elements are $S_{ij}=\frac{1}{2}(H_{ij}+H_{ji})$.
- From this it follows that $$p(S)=p(S_{11},\dots,S_{NN})=\prod_{i=1}^N\left(\frac{1}{\sqrt{2\pi}}\exp\left[-\frac{S_{ii}^2}{2}\right]\right)\prod_{1\leq i<j\leq N}\left(\frac{1}{\sqrt{\pi}}\exp\left[-S_{ij}^2\right]\right).$$
I really don't see how (1.) and (2.) imply (3.). So far I've tried writing $p(H)=\prod_{i=1}^N \prod_{i<j} \prod_{j<i}$ as a product of the main diagonal, upper triangle and lower triangle terms and swapping indices, but this runs into trouble because $S_{ij}^2=\frac{1}{4}(H_{ij}^2+H_{ji}^2+2H_{ij}H_{ji})$ has a cross term. I also tried using the change of variable formula (ie Jacobian) but this does not seem appropriate since there are $\frac{N(N+1)}{2}$ independent variables for $H_s$ but $N^2$ for $H$.
If anyone has any hints on how to complete this, I would greatly appreciate it!