The Symmetric Matrices mathrmSym(n)

# The $n\times n$ symmetric matrices $\mathrm{Sym}(n)$ embedded in $\mathbb R^{n\times n}$

The manifold of symmetric matrices $\mathcal{Sym}(n)$ posesses the following instances of the abstract types Manifold, MPoint, and TVector.

SymPoint <: MPoint

A point $x$ on the manifold $\mathcal M = \mathrm{Sym}(n)$ of $n\times n$ symmetric matrices, represented in the redundant way of a symmetric matrix (instead of storing just the upper half).

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SymTVector <: TVector

A tangent vector $\xi$ in $T_x\mathcal M$ of a symmetric matrix $x\in\mathcal M$.

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Symmetric <: Manifold

The manifold $\mathcal M = \mathrm{Sym}(n)$, where $\mathrm{Sym}(n) = \{ x \in \mathbb R^{n\times n} | x = x^\mathrm{T} \}$, $n\in\mathbb N$, denotes the manifold of symmetric matrices equipped with the trace inner product and its induced Forbenius norm.

Abbreviation

Sym or Sym(n), respectively.

Constructor

Symmetric(n)

generates the manifold of n-by-n symmetric matrices.

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## Functions

exp(M,x,ξ[, t=1.0])

compute the exponential map on the Symmetric manifold M given a SymPoint x and a SymTVector ξ, as well as an optional scaling factor t. The exponential map is given by

$\exp_{x}ξ = x+ξ.$
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log(M,x,y)

compute the logarithmic map for two SymPointx,y on the Symmetric M, which is given by $\log_xy = y-x$.

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dot(M,x,ξ,ν)

inner product of two SymTVectors ξ,ν lying in the tangent space of the SymPoint x on the Symmetric manifold M.

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norm(M,x,ξ)

computes the norm of the SymTVector ξ in the tangent space of the SymPoint x on the Symmetric M embedded in the Euclidean space, i.e. by its Frobenius norm.

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distance(M,x,y)

distance of two SymPoints x,y on the Symmetric manifold M inherited from embedding them in $\mathbb R^{n\times n}$, i.e. use the Frobenious norm of the difference.

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manifoldDimension(x)

returns the manifold dimension the SymPoint x belongs to.

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manifoldDimension(M)

returns the manifold dimension of the Symmetric manifold M.

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parallelTransport(M,x,y,ξ)

compute the parallel transport of a SymTVector ξ from the tangent space at the SymPoint x to the SymPointy on the Symmetric manifold M. Since the metric is inherited from the embedding space, it is just the identity.

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typicalDistance(M)

returns the typical distance on the Symmetric manifold M, i.e. $\sqrt{n}$.

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validateMPoint(M,x)

validate, that the SymPoint x is a valid point on the Symmetric manifold M, i.e. that its dimensions are correct and that the matrix is symmetric.

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validateTVector(M,x,ξ)

validate, that the SymTVector is a valid tangent vector to the SymPoint x on the Symmetric manifold M, i.e. that its dimensions are correct and that the matrix is symmetric.

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ξ = zeroTVector(M,x)

returns a zero vector in the tangent space $T_x\mathcal M$ of the SymPoint x on the Symmetric manifold M`.

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