Centered matrices

Manifolds.CenteredMatrices — Type

CenteredMatrices{m,n,𝔽} <: AbstractEmbeddedManifold{𝔽,TransparentIsometricEmbedding}

The manifold of $m × n$ real-valued or complex-valued matrices whose columns sum to zero, i.e.

\[\bigl\{ p ∈ 𝔽^{m × n}\ \big|\ [1 … 1] * p = [0 … 0] \bigr\},\]

where $𝔽 ∈ \{ℝ,ℂ\}$.

Constructor

CenteredMatrices(m, n[, field=ℝ])

Generate the manifold of m-by-n (field-valued) matrices whose columns sum to zero.

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ManifoldsBase.check_manifold_point — Method

check_manifold_point(M::CenteredMatrices{m,n,𝔽}, p; kwargs...)

Check whether the matrix is a valid point on the CenteredMatrices M, i.e. is an m-by-n matrix whose columns sum to zero.

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

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ManifoldsBase.check_tangent_vector — Method

check_tangent_vector(M::CenteredMatrices{m,n,𝔽}, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to manifold point p on the CenteredMatrices M, i.e. that X is a matrix of size (m,n) whose columns sum to zero and its values are from the correct AbstractNumbers. The optional parameter check_base_point indicates, whether to call check_manifold_point for p. The tolerance for the column sums of p and X can be set using kwargs....

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ManifoldsBase.manifold_dimension — Method

manifold_dimension(M::CenteredMatrices{m,n,𝔽})

Return the manifold dimension of the CenteredMatrices m-by-n matrix M over the number system 𝔽, i.e.

\[\dim(\mathcal M) = (m*n - n) \dim_ℝ 𝔽,\]

where $\dim_ℝ 𝔽$ is the real_dimension of 𝔽.

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ManifoldsBase.project — Method

project(M::CenteredMatrices, p, X)

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

\[\operatorname{proj}_p(X) = X - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m x_{j,i}$ for $i = 1, \dots, n$.

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ManifoldsBase.project — Method

project(M::CenteredMatrices, p)

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

\[\operatorname{proj}_{\mathcal M}(p) = p - \begin{bmatrix} 1\\ ⋮\\ 1 \end{bmatrix} * [c_1 \dots c_n],\]

where $c_i = \frac{1}{m}\sum_{j=1}^m p_{j,i}$ for $i = 1, \dots, n$.

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