# Centered matrices

Manifolds.CenteredMatrices โ Type
CenteredMatrices{T,๐ฝ} <: AbstractDecoratorManifold{๐ฝ}

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

$$$\bigl\{ p โ ๐ฝ^{mรn}\ \big|\ [1 โฆ 1] * p = [0 โฆ 0] \bigr\},$$$

where $๐ฝ โ \{โ,โ\}$.

Constructor

CenteredMatrices(m, n[, field=โ]; parameter::Symbol=:type)

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

parameter: whether a type parameter should be used to store m and n. By default size is stored in type. Value can either be :field or :type.

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ManifoldsBase.Weingarten โ Method
Y = Weingarten(M::CenteredMatrices, p, X, V)
Weingarten!(M::CenteredMatrices, Y, p, X, V)

Compute the Weingarten map $\mathcal W_p$ at p on the CenteredMatrices 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.

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ManifoldsBase.check_point โ Method
check_point(M::CenteredMatrices, 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_vector โ Method
check_vector(M::CenteredMatrices, p, X; 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 tolerance for the column sums of p and X can be set using kwargs....

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ManifoldsBase.manifold_dimension โ Method
manifold_dimension(M::CenteredMatrices)

Return the manifold dimension of the CenteredMatrices m-by-n matrix M over the number system ๐ฝ, i.e.

$$$\dim(\mathcal M) = (m*n - n) \dim_โ ๐ฝ,$$$

where $\dim_โ ๐ฝ$ is the real_dimension of ๐ฝ.

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ManifoldsBase.project โ Method
project(M::CenteredMatrices, p, X)

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

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

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

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ManifoldsBase.project โ Method
project(M::CenteredMatrices, p)

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

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

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

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