symplecticstiefel.md

(Real) Symplectic Stiefel

The SymplecticStiefel manifold, denoted \operatorname{SpSt}(2n, 2k), represents canonical symplectic bases of 2k dimensonal symplectic subspaces of ℝ^{2n×2n}. This means that the columns of each element p \in \operatorname{SpSt}(2n, 2k) \subset ℝ^{2n×2k} constitute a canonical symplectic basis of \operatorname{span}(p). The canonical symplectic form is a non-degenerate, bilinear, and skew symmetric map \omega_{2k}\colon 𝔽^{2k}×𝔽^{2k} → 𝔽, given by \omega_{2k}(x, y) = x^T Q_{2k} y for elements x, y \in 𝔽^{2k}, with

$$Q_{2k} = \begin{bmatrix} 0_k & I_k \\\ -I_k & 0_k \end{bmatrix}.$$

Specifically given an element p \in \operatorname{SpSt}(2n, 2k) we require that

$$\omega_{2n} (p x, p y) = x^T(p^TQ_{2n}p)y = x^TQ_{2k}y = \omega_{2k}(x, y) \;\forall\; x, y \in 𝔽^{2k},$$

leading to the requirement on p that p^TQ_{2n}p = Q_{2k}. In the case that k = n, this manifold reduces to the SymplecticMatrices manifold, which is also known as the symplectic group.

Modules = [Manifolds]
Pages = ["manifolds/SymplecticStiefel.jl"]
Order = [:type, :function]

Literature

Pages = ["symplecticstiefel.md"]
Canonical=false