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We investigate traces of powers of random matrices whose distributions areinvariant under rotations (with respect to the Hilbert--Schmidt inner product)within a real-linear subspace of the space of $n\times n$ matrices. Thematrices we consider may be real or complex, and Hermitian, antihermitian, orgeneral. We use Stein's method to prove multivariate central limit theorems,with convergence rates, for these traces of powers, which imply central limittheorems for polynomial linear eigenvalue statistics. In contrast to the usualsituation in random matrix theory, in our approach general, nonnormal matricesturn out to be easier to study than Hermitian matrices.

Fluctuations of the spectrum in rotationally invariant random matrix ensembles