  Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, there
is a canonical probability distribution on the set of real symmetric or complex
Hermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In this
paper, we study various features of random matrices with this distribution. Our
main results show that under mild conditions, when $n$ is large, linear
functionals of the entries of such random matrices have approximately Gaussian
joint distributions. The results take the form of upper bounds on distances
between multivariate distributions, which allows us also to consider the case
when the number of linear functionals grows with $n$. In the context of quantum
mechanics, these results can be viewed as describing the joint probability
distribution of the expectation values of a family of observables on a quantum
system in a random mixed state. Other applications are given to spectral
distributions of submatrices, the classical invariant ensembles, and to a
probabilistic counterpart of the Schur--Horn theorem, relating eigenvalues and
diagonal entries of Hermitian matrices.
