The difference between the Grassmannian manifold and Stiefel Manifold.

[Muntadher Q Alsabah]

Hi dears,

Could you please explain to me what is/are the difference between the Grassmannian manifold and Stiefel Manifold ? Both are complex. 

Appreciate your help.

Best reagrds

[Nicolas Boumal]

Hello Muntadher,

The (non-compact) Stiefel manifold St(n, p) is the set of matrices of size n x p whose columns are orthonormal.

The Grassmann manifold is an abstract manifold: you can think of it as the quotient of the Stiefel manifold by the equivalence relation that says: two matrices are equivalent if their columns span the same subspace. In practice, Manopt represents points on the Grassmann manifold the same way it does points on the Stiefel manifold: as matrices with orthonormal columns. But the underlying geometry is different, hence so are a number of operations (such as orthogonal projection to the tangent space or horizontal space.)

I encourage you to read up in Absil et al.'s book.

Best,

Nicolas

[Muntadher Q Alsabah]

Dear Nicolas,

I will have a deep look at the book. Thank you very much for your suggestion. 

Best regards,

Muntadher