Invertible matrices

InvertibleMatrices{𝔽,T} <: AbstractDecoratorManifold{𝔽}

The AbstractManifold consisting of the real- or complex-valued invertible matrices, that is the set

\[\bigl\{p ∈ 𝔽^{n×n}\ \big|\ \det(p) \neq 0 \bigr\},\]

where the field $𝔽 ∈ \{ ℝ, ℂ\}$.

Constructor

InvertibleMatrices(n::Int, field::AbstractNumbers=ℝ)

Generate the manifold of $n×n$ invertible matrices.

Random.rand(M::InvertibleMatrices; vector_at=nothing, kwargs...)

If vector_at is nothing, return a random point on the InvertibleMatrices manifold M by using rand in the embedding.

If vector_at is not nothing, return a random tangent vector from the tangent space of the point vector_at on the InvertibleMatrices by using by using rand in the embedding.

Y = Weingarten(M::InvertibleMatrices, p, X, V)
Weingarten!(M::InvertibleMatrices, Y, p, X, V)

Compute the Weingarten map $\mathcal W_p$ at p on the InvertibleMatrices M with respect to the tangent vector $X \in T_p\mathcal M$ and the normal vector $V \in N_p\mathcal M$.

Since this a flat space by itself, the result is always the zero tangent vector.

check_point(M::InvertibleMatrices{n,𝔽}, p; kwargs...)

Check whether p is a valid manifold point on the InvertibleMatrices M, i.e. whether p is an invertible matrix of size (n,n) with values from the corresponding AbstractNumbers 𝔽.

check_vector(M::InvertibleMatrices{n,𝔽}, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the InvertibleMatrices M, which are all matrices of size $n×n$ its values have to be from the correct AbstractNumbers.

is_flat(::InvertibleMatrices)

Return true. InvertibleMatrices is a flat manifold.

manifold_dimension(M::InvertibleMatrices{n,𝔽})

Return the dimension of the InvertibleMatrices matrix M over the number system 𝔽, which is the same dimension as its embedding, the Euclidean(n, n; field=𝔽).