= Gaussian orthogonal ensemble
{c}
{title2=$\mathrm{GOE}(n)$}
= GOE
{c}
{synonym}
In the unscaled variance-one convention, a Gaussian orthogonal ensemble <matrix> $M$ is a real <symmetric matrix> whose upper-triangular entries are <independent>, with $M_{ij}\sim N(0,1)$ for $i<j$ and $M_{ii}\sim N(0,2)$. Its <joint probability density function> relative to the upper-triangular coordinates is proportional to $\exp[-\operatorname{tr}(M^2)/4]$. The scaled convention uses $M/\sqrt n$; the scaling must be stated. If $X$ has independent standard <normal> entries, $(X+X^T)/\sqrt2$ realizes the unscaled convention. Expanding in the entries of $X$ gives the <covariance> identity $\mathbb E[M_{ij}M_{kl}]=\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}$, with <Kronecker deltas>.
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