General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions

We study a random positive definite symmetric matrix distributed according to a real Wishart distribution. We compute general moments of the random matrix and of its inverse explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study of Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.