Matrices of determinant one

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

The AbstractManifold consisting of the real- or complex-valued (invertible) matrices od determinant one, that is the set

\[\bigl\{p ∈ 𝔽^{n×n}\ \big|\ \det(p) = 1 \bigr\},\]

where the field $𝔽 ∈ \{ ℝ, ℂ\}$.

Note that this is a subset of InvertibleMatrices, and a superset of any of the GeneralUnitaryMatrices

The tangent space at any point p is the set of matrices with trace 0.

Constructor

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

Generate the manifold of $n×n$ matrices of determinant one.

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

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

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

Check whether p is a valid manifold point on the DeterminantOneMatrices M, i.e. whether p has a determinant of $1$.

The check is perfomed with isapprox and all keyword arguments are passed to this

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

Check whether X is a tangent vector to manifold point p on the DeterminantOneMatrices M, which are all matrices of size $n×n$ with trace 0.

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

Return the dimension of the DeterminantOneMatrices matrix M over the number system 𝔽, which is one dimension less than its embedding, the Euclidean(n, n; field=𝔽).

project(G::DeterminantOneMatrices, p, X)
project!(G::DeterminantOneMatrices, Y, p, X)

Orthogonally project $X ∈ 𝔽^{n×n}$ onto the tangent space of $p$ to the DeterminantOneMatrices.

This first changes the representation from X to the trace-zero component, i.e. computes Y = p \ X and then subtracts c = tr(Y) / n from all diagonal entries.

project(G::DeterminantOneMatrices, p)
project!(G::DeterminantOneMatrices, q, p)

Project $p ∈ \mathrm{GL}(n, 𝔽)$ to the DeterminantOneMatrices using the singular value decomposition of $p = U S V^\mathrm{H}$.

The formula for the projection is

\[\operatorname{proj}(p) = U S D V^\mathrm{H},\]

where

\[D_{ij} = δ_{ij} \begin{cases} 1 & \text{ if } i ≠ n \\ \det(p)^{-1} & \text{ if } i = n \end{cases}.\]

The operation can be done in-place of q.