I am interested in generating random members of two classes of $n \times n$ positive-definite matrices $A$ and $B$--the former symmetric in nature, the latter, Hermitian.
The standard (Ginibre-matrix-based) approach (Ginibre) to obtaining random realizations of $A$ is to generate an $n \times n$ matrix $a$, the $n^2$ entries of which are standard univariate normal variates. Then, the desired realization of $A$ is gotten by computing the matrix product $a a^T$, where the transpose is indicated.
To obtain random realizations of $B$, one starts with an $n \times n$ matrix $b$, the $n^2$ entries of which are of the form $c + d I$, where $c$ and $d$ are themselves standard univariate normal variates. Then, the desired realization of $B$ is obtained by computing the matrix product $b b^\dagger$, where the conjugate transpose is denoted.
(If the random realizations of $A$ and $B$ are normalized to have trace 1, then they are--in physics parlance--density matrices randomly distributed with respect to Hilbert-Schmidte measure. It is, then, of interest to test their "partial transposes" for positive-definiteness. For $n=4,6$, such tests are equivalent to ones for "separability" [PPT].)
Can such desired random realizations be accomplished with the WishartMatrixDistribution command (in conjunction with the RandomVariate command), and, if so, might there be an associated speed-up in comparison to the Ginibre-matrix-product approach outlined above?
Also, more generally, what if the classes of matrices are $k \times n$, with $k<n$? (This pertains to the PPT/separability problem for $n \times n$ matrices of rank $k$.)