  We study the geometry of flag manifolds under different embeddings into a
product of Grassmannians. We show that differential geometric objects and
operations -- tangent vector, metric, normal vector, exponential map, geodesic,
parallel transport, gradient, Hessian, etc -- have closed-form analytic
expressions that are computable with standard numerical linear algebra.
Furthermore, we are able to derive a coordinate descent method in the flag
manifold that performs well compared to other gradient descent methods.
