Stiefel manifold

The Stiefel manifold of k-frames in n-dimensional real space is the set of all orthonormal k-frames, that is, ordered k-tuples of mutually perpendicular unit vectors; it is named after the Swiss mathematician Eduard Stiefel. Complex and quaternionic Stiefel manifolds are defined the same way using orthonormal frames in complex and quaternionic space, and the construction applies to any real, complex or quaternionic inner product space. Writing a frame as a matrix A whose columns are the vectors, the orthonormality condition is that the conjugate transpose of A times A is the identity matrix, and the resulting space is a compact manifold; in the real case its dimension is nk minus k(k+1)/2. Each Stiefel manifold is a homogeneous space: the orthogonal group acts transitively on real frames, so the real Stiefel manifold is the quotient of O(n) by O(n-k), and the unitary and symplectic groups act in the same way in the complex and quaternionic cases. Special cases include the unit sphere when k is 1, the unit tangent bundle of a sphere when k is 2, and the orthogonal group itself when k equals n. A uniform invariant measure can be sampled by applying the QR factorization to a Gaussian random matrix.