The Bayesian analysis of a state-space model includes computing the posterior distribution of the system's parameters as well as its latent states. When the latent states wander around R n there are several well-known modeling components and computational tools that may be protably combined to achieve this task. When the latent states are constrained to a strict subset of R n these models and tools are either impaired or break down completely. State-space models whose latent states are covariance matrices arise in nance and exemplify the challenge of devising tractable models in the constrained setting. To that end, we present a state-space model whose observations and latent states take values on the manifold of symmetric positive-denite matrices and for which one may easily compute the posterior distribution of the latent states and the system's parameters as well as ltered distributions and one-step ahead predictions. Employing the model within the context of nance, we show how one can use realized covariance matrices as data to predict latent time-varying covariance matrices. This approach out-performs factor stochastic volatility.