Integration of invariant matrices and application to statistics

We consider random matrices that have invariance properties under the action of unitary groups (either a left-right invariance, or a conjugacy invariance), and we give formulas for moments in terms of functions of eigenvalues. Our main tool is the Weingarten calculus. As an application to statistics, we obtain new formulas for the pseudo inverse of Gaussian matrices and for the inverse of compound Wishart matrices.