Unit-norm symmetric matrices

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

The AbstractManifold consisting of the $n×n$ symmetric matrices of unit Frobenius norm, i.e.

\[\mathcal{S}_{\text{sym}} :=\bigl\{p ∈ 𝔽^{n×n}\ \big|\ p^{\mathrm{H}} = p, \lVert p \rVert = 1 \bigr\},\]

where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose, and the field $𝔽 ∈ \{ ℝ, ℂ\}$.

Constructor

SphereSymmetricMatrices(n[, field=ℝ]; parameter::Symbol=:type)

Generate the manifold of n-by-n symmetric matrices of unit Frobenius norm.

check_point(M::SphereSymmetricMatrices, p; kwargs...)

Check whether the matrix is a valid point on the SphereSymmetricMatrices M, i.e. is an n-by-n symmetric matrix of unit Frobenius norm.

The tolerance for the symmetry of p can be set using kwargs....

check_vector(M::SphereSymmetricMatrices, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the SphereSymmetricMatrices M. That is, X has to be a symmetric matrix of size (n,n) that is orthogonal to p with respect to the Frobenius inner product.

The tolerance for the symmetry of p and X can be set using kwargs....

is_flat(::SphereSymmetricMatrices)

Return false. SphereSymmetricMatrices is not a flat manifold.

manifold_dimension(M::SphereSymmetricMatrices{𝔽})

Return the manifold dimension of the SphereSymmetricMatrices n-by-n symmetric matrix M of unit Frobenius norm over the number system 𝔽, i.e.

\[\begin{aligned} \dim(\mathcal{S}_{\text{sym}})(n,ℝ) &= \frac{n(n+1)}{2} - 1,\\ \dim(\mathcal{S}_{\text{sym}})(n,ℂ) &= 2\frac{n(n+1)}{2} - n -1. \end{aligned}\]

project(M::SphereSymmetricMatrices, p, X)

Project the matrix X onto the tangent space at p on the SphereSymmetricMatrices M, i.e.

\[\operatorname{proj}_p(X) = \frac{X + X^{\mathrm{H}}}{2} - ⟨p, \frac{X + X^{\mathrm{H}}}{2}⟩p,\]

where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transposed.

project(M::SphereSymmetricMatrices, p)

Projects p from the embedding onto the SphereSymmetricMatrices M, i.e.

\[\operatorname{proj}_{\mathcal{S}_{\text{sym}}}(p) = \frac{s}{\lVert s \rVert}, \qquad\text{where } s = \frac{1}{2} \bigl( p + p^{\mathrm{H}} \bigr),\]

where $⋅^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transposed, and $\lVert ⋅ \rVert$ denotes the Frobenius norm.