On some known derivations and new ones for the Wishart distribution: A didactic

The proofs of the probability density function (pdf) of the Wishart distribution tend to be complicated with geometric viewpoints, tedious Jacobians and not self-contained algebra. In this paper, some known proofs and simple new ones for uncorrelated and correlated cases are provided with didactic explanations. For the new derivation of the uncorrelated case, an elementary direct derivation of the distribution of the Bartlett-decomposed matrix is provided. In the derivation of the correlated case from the uncorrelated one, simple methods including a new one are shown.