Wishart distribution

Let $U_i\sim N_p({\mu }_i,\mathrm{\Sigma }),i=1,\mathrm{\dots },k$ be independent $p$-dimensional random variables, which aremultivariate normally distributed (http://planetmath.org/jointnormaldistribution).Let $S={\sum }_{i=1}^k{U}_{i}U_i{}^T$. Let $M$ be the $k\times p$ matrixwith ${\mu }_1,\mathrm{\dots },{\mu }_k$ as rows.Then the joint distribution of theelements of $S$ is said to be a Wishart distribution on $k$ of freedom , and isdenoted by $W_p(k,\mathrm{\Sigma },M)$. If $M=0$, the distribution is said to becentral and is denoted by $W_p(k,\mathrm{\Sigma })$.The Wishart distribution is a multivariate generalization of the ${\chi }^2$ distribution.

$W_p$ has a density function when $k\ge p$.