  Metrics on Grassmannians have a wide array of applications: machine learning,
wireless communication, computer vision, etc. But the available distances
between subspaces of distinct dimensions present problems, and the dimensional
asymmetry of the subspaces calls for the use of asymmetric metrics. We extend
the Fubini-Study metric as an asymmetric angle with useful properties, and
whose relations to products of Grassmann and Clifford geometric algebras make
it easy to compute. We also describe related angles that provide extra
information, and a method to extend other Grassmannian metrics to asymmetric
metrics on the full Grassmannian.
