Hamiltonian matrices

Hamiltonian{T,S<:AbstractMatrix{<:T}} <: AbstractMatrix{T}

A type to store a Hamiltonian matrix, that is a square matrix for which $A^+ = -A$ where

\[A^+ = J_{2n}^{\mathrm{T}}A^{\mathrm{T}}J_{2n}, \qquad J_{2n} = \begin{pmatrix} 0 & I_n\\-I_n & 0 \end{pmatrix},\]

and $I_n$ denotes the $n×n$ identity matrix.

HamiltonianMatrices{𝔽, T} <: AbstractDecoratorManifold{𝔽}

The AbstractManifold consisting of Hamiltonian matrices of size $2n×2n$, i.e. the set

\[\mathfrak{sp}(2n,𝔽) = \bigl\{p ∈ 𝔽^{2n×2n}\ \big|\ p^+ = -p \bigr\},\]

where $⋅^{+}$ denotes the symplectic_inverse, and $𝔽 ∈ \{ ℝ, ℂ\}$.

Though it is slightly redundant, usually the matrices are stored as $2n×2n$ arrays.

The symbol $\mathfrak{sp}$ refers to the main usage within Manifolds.jl that is the Lie algebra to the SymplecticMatrices interpreted as a Lie group with the matrix multiplication as group operation.

Constructor

HamiltonianMatrices(2n::Int, field::AbstractNumbers=ℝ; parameter::Symbol=:type)

Generate the manifold of $2n×2n$ Hamiltonian matrices.

^(A::Hamiltonian, ::typeof(+))

Compute the symplectic_inverse of a Hamiltonian (A)

pX = rand(M::HamiltonianMatrices; σ::Real=1.0, vector_at=nothing)
rand!(rng::AbstractRNG, M::HamiltonianMatrices, pX; σ::Real=1.0, vector_at=nothing)

Generate a random Hamiltonian matrix. Since these are a submanifold of $ℝ^{2n×2n}$, the same method applies for points and tangent vectors. This can also be done in-place of pX.

The construction is based on generating one normally-distributed $n×n$ matrix $A$ and two symmetric $n×n$ matrices $B, C$ which are then stacked:

\[p = \begin{pmatrix} A & B\\ C & -A^{\mathrm{T}} \end{pmatrix}\]

is_hamiltonian(A::AbstractMatrix; kwargs...)

Test whether a matrix A is hamiltonian. The test consists of verifying whether

\[A^+ = -A\]

where $A^+$ denotes the symplectic_inverse of A.

The passed keyword arguments are passed on to isapprox check within

check_point(M::HamiltonianMatrices, p; kwargs...)

Check whether p is a valid manifold point on the HamiltonianMatrices M, i.e. whether p is_hamiltonian.

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

check_vector(M::HamiltonianMatrices, p, X; kwargs... )

Check whether X is a tangent vector to manifold point p on the HamiltonianMatrices M, i.e. X has to be a Hamiltonian matrix The tolerance for is_hamiltonian X can be set using kwargs....

is_flat(::HamiltonianMatrices)

Return true. HamiltonianMatrices is a flat manifold.

manifold_dimension(M::HamiltonianMatrices)

Return the dimension of the HamiltonianMatrices M over the number system 𝔽, that is

\[\dim \mathfrak{sp}(2n,𝔽) = (2n+1)n \dim_ℝ 𝔽,\]

where $\dim_ℝ 𝔽$ is the real_dimension of $𝔽$. For $𝔽 = ℝ$ this is the dimension of the SymplecticMatrices $\mathrm{Sp}(2n,ℝ)$, whose Lie algebra this is.

project(::HamiltonianMatrices, p, X)
project!(::HamiltonianMatrices, Y, p, X)

Project a matrix $X ∈ 𝔽^{2n×2n}$ onto the tangent space $T_p\mathfrak{sp}(2n,𝔽)$ of the HamiltonianMatrices at $p$, which is $\mathfrak{sp}(2n,𝔽)$ itself, so that $\operatorname{proj}_p(X) = \frac{1}{2}\bigl( X - X^{+} \bigr)$ as for points.

project(::HamiltonianMatrices, p)
project!(::HamiltonianMatrices, q, p)

Project a matrix $p ∈ 𝔽^{2n×2n}$ onto the HamiltonianMatrices $\mathfrak{sp}(2n,𝔽)$ relative to the Euclidean metric of the embedding $𝔽^{2n×2n}$, that is

\[ \operatorname{proj}_{\mathfrak{sp}(2n,𝔽)}(p) = \frac{1}{2}\bigl( p - p^{+} \bigr),\]

where $p^+$ denotes the symplectic_inverse of $p$. For $𝔽 = ℝ$ this is Proposition 2 in [GSAS21a] at the identity, whose tangent space is $\mathfrak{sp}(2n,ℝ)$ by Corollary 3.46 in [Hal15]. For $𝔽 = ℂ$ the real and the imaginary part are projected separately, see Proposition 3.38 in [Hal15].

riemann_tensor(M::HamiltonianMatrices, p, X, Y, Z)

Compute the Riemann tensor $R(X,Y)Z$ at point p on HamiltonianMatrices M. Its value is always the zero tangent vector, since M is flat.

sectional_curvature(M::HamiltonianMatrices, p, X, Y)

Return the sectional curvature 0 of HamiltonianMatrices M, which is flat.

sectional_curvature_max(M::HamiltonianMatrices)

Return the upper bound 0 of the sectional curvature of HamiltonianMatrices M, which is flat.

sectional_curvature_min(M::HamiltonianMatrices)

Return the lower bound 0 of the sectional curvature of HamiltonianMatrices M, which is flat.

pX = rand(M::HamiltonianMatrices; σ::Real=1.0, vector_at=nothing)
rand!(rng::AbstractRNG, M::HamiltonianMatrices, pX; σ::Real=1.0, vector_at=nothing)

Generate a random Hamiltonian matrix. Since these are a submanifold of $ℝ^{2n×2n}$, the same method applies for points and tangent vectors. This can also be done in-place of pX.

The construction is based on generating one normally-distributed $n×n$ matrix $A$ and two symmetric $n×n$ matrices $B, C$ which are then stacked:

\[p = \begin{pmatrix} A & B\\ C & -A^{\mathrm{T}} \end{pmatrix}\]