  We extend the application of Hamiltonian Monte Carlo to allow for sampling
from probability distributions defined over symmetric or Hermitian positive
definite matrices. To do so, we exploit the Riemannian structure induced by
Cartan's century-old canonical metric. The geodesics that correspond to this
metric are available in closed-form and---within the context of Lagrangian
Monte Carlo---provide a principled way to travel around the space of positive
definite matrices. Our method improves Bayesian inference on such matrices by
allowing for a broad range of priors, so we are not limited to conjugate priors
only. In the context of spectral density estimation, we use the (non-conjugate)
complex reference prior as an example modeling option made available by the
algorithm. Results based on simulated and real-world multivariate time series
are presented in this context, and future directions are outlined.
