We introduce a Brownian motion model for the parametric evolution of eigenstates of a complex quantum system, modelled by a random matrix. The model is analogous to Dyson's model for the evolution of the eigenvalues. We use this approach to analyse correlation functions describing the parameter dependence of diagonal and off-diagonal matrix elements of a generic operator. In the case of diagonal matrix elements, we compare our results with a semiclassical approach, which relates sums of matrix elements to periodic classical orbits. For systems with a chaotic classical limit, the semiclassical correlation function agrees exactly with the random matrix theory.