Unit-norm symmetric matrices

Manifolds.SphereSymmetricMatrices โ€” Type
SphereSymmetricMatrices{n,๐”ฝ} <: AbstractEmbeddedManifold{โ„,TransparentIsometricEmbedding}

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

\[\mathcal{S}_{\text{sym}} :=\bigl\{p โˆˆ ๐”ฝ^{n ร— n}\ \big|\ p^{\mathrm{H}} = p, \lVert p \rVert = 1 \bigr\},\]

where $\cdot^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transpose, and the field $๐”ฝ โˆˆ \{ โ„, โ„‚\}$.

Constructor

SphereSymmetricMatrices(n[, field=โ„])

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

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ManifoldsBase.check_point โ€” Method
check_point(M::SphereSymmetricMatrices{n,๐”ฝ}, 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....

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ManifoldsBase.check_vector โ€” Method
check_vector(M::SphereSymmetricMatrices{n,๐”ฝ}, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the SphereSymmetricMatrices M, i.e. X has to be a symmetric matrix of size (n,n) of unit Frobenius norm.

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

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ManifoldsBase.manifold_dimension โ€” Method
manifold_dimension(M::SphereSymmetricMatrices{n,๐”ฝ})

Return the manifold dimension of the SphereSymmetricMatrices n-by-n symmetric matrix M of unit Frobenius norm over the number system ๐”ฝ, i.e.

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

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ManifoldsBase.project โ€” Method
project(M::SphereSymmetricMatrices, p, X)

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

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

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

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ManifoldsBase.project โ€” Method
project(M::SphereSymmetricMatrices, p)

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

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

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

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