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 wit h X and Y independent, mean 0 and variance 1/2 normal variables. We sh ow that for j = 1, 2, ..., Tr(M(j)) are independent and distributed as square-root jZ asymptotically as n --> infinity. This result is used to study the set of eigenvalues of M, Similar results are given for th e orthogonal and symplectic and symmetric groups.