On the eigenvalues of random matrices

Let M be a random matrix chosen from Haar measure on the unitary group U n . Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr( M j ) are independent and distributed as √ jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.