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

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

and $I_n$ denotes the $n×n$

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

The AbstractManifold consisting of (real-valued) hamiltonian matrices of size $n×n$, 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 $\mathfak{sp}$ 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)

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!(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.