  We derive three families of orthogonally-equivariant matrix submanifold
models for the Grassmann, flag, and Stiefel manifolds respectively. These
families are exhaustive -- every orthogonally-equivariant submanifold model of
the lowest dimension for any of these manifolds is necessarily a member of the
respective family, with a small number of exceptions. They have several
computationally desirable features. The orthogonal equivariance allows one to
obtain, for various differential geometric objects and operations, closed-form
analytic expressions that are readily computable with standard numerical linear
algebra. The minimal dimension aspect translates directly to a speed advantage
in computations. And having an exhaustive list of all possible matrix models
permits one to identify the model with the lowest matrix condition number,
which translates to an accuracy advantage in computations. As an interesting
aside, we will see that the family of models for the Stiefel manifold is
naturally parameterized by the Cartan manifold, i.e., the positive definite
cone equipped with its natural Riemannian metric.
