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Let $W$ be a random positive definite symmetric matrix distributed accordingto a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inversematrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ...W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingartenfunction, which was recently introduced in the study for Haar-distributedorthogonal matrices. As applications, we give formulas for moments of traces ofa Wishart matrix and its inverse.

General moments of the inverse real Wishart distribution and orthogonal Weingarten functions