= Gaussian unitary ensemble
{c}
{title2=$\mathrm{GUE}(n)$}
= GUE
{c}
{synonym}
In the unscaled convention with diagonal <variance> one, a Gaussian unitary ensemble <matrix> $H$ is a <Hermitian matrix> with independent real diagonal entries $N(0,1)$ and upper off-diagonal entries $(A_{ij}+iB_{ij})/\sqrt2$, where all the $A_{ij},B_{ij}$ and diagonal entries are <independent> and have the <standard normal distribution>. Its density on the real coordinates of <Hermitian matrices> is proportional to $\exp[-\operatorname{tr}(H^2)/2]$. Its exchangeably labelled real <eigenvalues> have symmetric <joint probability density function> proportional to $\prod_{i<j}(\lambda_i-\lambda_j)^2e^{-\sum_i\lambda_i^2/2}$. The squared <Vandermonde determinant> suppresses coincident <eigenvalues>; <orthogonal polynomial projection kernels> express the <correlation functions of a point process> as <determinants>.
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