This paper presents a geometric approach to estimating subspaces as
elements of complex Grassmann-manifold, with each subspace
represented by its unique, complex projection matrix.
Variation between the subspaces is modeled by rotating their projection
matrices
via the action of unitary matrices (elements of the unitary group (${\bf U}(n)$)).
Subspace estimation or tracking then corresponds to the inferences on
${\bf U}(n)$.
Taking a Bayesian approach, a posterior density is derived on
${\bf U}(n)$ and certain expectations  under this posterior are empirically generated.
For the choice of Hilbert-Schmidt
norm on ${\bf U}(n)$, to define estimation errors,
an optimal MMSE estimator is derived.
It is shown that this estimator achieves a lower bound,
defined on the expected squared-errors associated with all possible estimators.
The estimator and the bound are  computed using (Metropolis-Adjusted)
Langevin's-diffusion algorithm for sampling from the posterior.
For use in subspace tracking
a prior model on subspace rotation, that utilizes Newtonian dynamics, is suggested.