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Concept
Stiefel manifold
Summary
In mathematics, the Stiefel manifold is the set of all orthonormal k-frames in That is, it is the set of ordered orthonormal k-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold of orthonormal k-frames in and the quaternionic Stiefel manifold of orthonormal k-frames in . More generally, the construction applies to any real, complex, or quaternionic inner product space.
In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in or this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group.
Let stand for or The Stiefel manifold can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in The orthonormality condition is expressed by AA = where A denotes the conjugate transpose of A and denotes the k × k identity matrix. We then have
The topology on is the subspace topology inherited from With this topology is a compact manifold whose dimension is given by
Each of the Stiefel manifolds can be viewed as a homogeneous space for the action of a classical group in a natural manner.
Every orthogonal transformation of a k-frame in results in another k-frame, and any two k-frames are related by some orthogonal transformation. In other words, the orthogonal group O(n) acts transitively on The stabilizer subgroup of a given frame is the subgroup isomorphic to O(n−k) which acts nontrivially on the orthogonal complement of the space spanned by that frame.
Likewise the unitary group U(n) acts transitively on with stabilizer subgroup U(n−k) and the symplectic group Sp(n) acts transitively on with stabilizer subgroup Sp(n−k).
In each case can be viewed as a homogeneous space:
When k = n, the corresponding action is free so that the Stiefel manifold is a principal homogeneous space for the corresponding classical group.
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