  We propose a novel Metropolis-Hastings algorithm to sample uniformly from the
space of correlation matrices. Existing methods in the literature are based on
elaborated representations of a correlation matrix, or on complex
parametrizations of it. By contrast, our method is intuitive and simple, based
the classical Cholesky factorization of a positive definite matrix and Markov
chain Monte Carlo theory. We perform a detailed convergence analysis of the
resulting Markov chain, and show how it benefits from fast convergence, both
theoretically and empirically. Furthermore, in numerical experiments our
algorithm is shown to be significantly faster than the current alternative
approaches, thanks to its simple yet principled approach.
