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