stiefel.md

Stiefel

Common and metric independent functions

Modules = [Manifolds]
Pages = ["manifolds/Stiefel.jl"]
Order = [:type, :function]

Default metric: the Euclidean metric

The [EuclideanMetric](@extref ManifoldsBase.EuclideanMetric) is obtained from the embedding of the Stiefel manifold in ℝ^{n,k}.

Modules = [Manifolds]
Pages = ["manifolds/StiefelEuclideanMetric.jl"]
Order = [:function]

The canonical metric

Any X∈T_p\mathcal M, p∈\mathcal M, can be written as

$$X = pA + (I_n-pp^{\mathrm{T}})B, \quad A ∈ ℝ^{p×p} \text{ skew-symmetric}, \quad B ∈ ℝ^{n×p} \text{ arbitrary.}$$

In the [EuclideanMetric](@extref ManifoldsBase.EuclideanMetric), the elements from A are counted twice (i.e. weighted with a factor of 2). The canonical metric avoids this.

Modules = [Manifolds]
Pages = ["manifolds/StiefelCanonicalMetric.jl"]
Order = [:type]

Modules = [Manifolds]
Pages = ["manifolds/StiefelCanonicalMetric.jl"]
Order = [:function]

The submersion or normal metric

Modules = [Manifolds]
Pages = ["manifolds/StiefelSubmersionMetric.jl"]
Order = [:type, :function]
Public = true
Private = false

Internal types and functions

Modules = [Manifolds]
Pages = ["manifolds/StiefelSubmersionMetric.jl"]
Order = [:type, :function]
Public = false
Private = true

Literature

Pages = ["stiefel.md"]
Canonical=false