Wishart and anti-Wishart random matrices

We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices A(dagger)A, for any finite number of rows and columns of A, without any large N approximations. In particular, we treat the case when the Wishart-type random matrix contains redundant, nonrandom information, which is a new result. This representation is of interest for a procedure for reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and artificial intelligence networks.