Symmetric matrices

SymmetricMatrices{n,𝔽} <: AbstractDecoratorManifold{𝔽}

The AbstractManifold $ \operatorname{Sym}(n)$ consisting of the real- or complex-valued symmetric matrices of size $n × n$, i.e. the set

\[\operatorname{Sym}(n) = \bigl\{p ∈ 𝔽^{n × n}\ \big|\ p^{\mathrm{H}} = p \bigr\},\]

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

Though it is slightly redundant, usually the matrices are stored as $n × n$ arrays.

Note that in this representation, the complex valued case has to have a real-valued diagonal, which is also reflected in the manifold_dimension.

Constructor

SymmetricMatrices(n::Int, field::AbstractNumbers=ℝ)

Generate the manifold of $n × n$ symmetric matrices.

Y = Weingarten(M::SymmetricMatrices, p, X, V)
Weingarten!(M::SymmetricMatrices, Y, p, X, V)

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

check_point(M::SymmetricMatrices{n,𝔽}, p; kwargs...)

Check whether p is a valid manifold point on the SymmetricMatrices M, i.e. whether p is a symmetric matrix of size (n,n) with values from the corresponding AbstractNumbers 𝔽.

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

check_vector(M::SymmetricMatrices{n,𝔽}, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the SymmetricMatrices M, i.e. X has to be a symmetric matrix of size (n,n) and its values have to be from the correct AbstractNumbers.

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

is_flat(::SymmetricMatrices)

Return true. SymmetricMatrices is a flat manifold.

manifold_dimension(M::SymmetricMatrices{n,𝔽})

Return the dimension of the SymmetricMatrices matrix M over the number system 𝔽, i.e.

\[\begin{aligned} \dim \mathrm{Sym}(n,ℝ) &= \frac{n(n+1)}{2},\\ \dim \mathrm{Sym}(n,ℂ) &= 2\frac{n(n+1)}{2} - n = n^2, \end{aligned}\]

where the last $-n$ is due to the zero imaginary part for Hermitian matrices

project(M::SymmetricMatrices, p, X)

Project the matrix X onto the tangent space at p on the SymmetricMatrices M,

\[\operatorname{proj}_p(X) = \frac{1}{2} \bigl( X + X^{\mathrm{H}} \bigr),\]

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

project(M::SymmetricMatrices, p)

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

\[\operatorname{proj}_{\operatorname{Sym}(n)}(p) = \frac{1}{2} \bigl( p + p^{\mathrm{H}} \bigr),\]

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