  In this paper, we present new results on the Riemannian geometry of symmetric
positive semi-definite (SPSD) matrices. First, based on an existing
approximation of the geodesic path, we introduce approximations of the
logarithmic and exponential maps. Second, we present a closed-form expression
for Parallel Transport (PT). Third, we derive a canonical representation for a
set of SPSD matrices. Based on these results, we propose an algorithm for
Domain Adaptation (DA) and demonstrate its performance in two applications:
fusion of hyper-spectral images and motion identification.
