On skew-Hamiltonian Matrices and their Krylov-Lagrangian Subspaces

It is a well-known fact that the Krylov space $\mathcal{K}_j(H,x)$ generated by a skew-Hamiltonian matrix $H \in \mathbb{R}^{2n \times 2n}$ and some $x \in \mathbb{R}^{2n}$ is isotropic for any $j \in \mathbb{N}$. For any given isotropic subspace $\mathcal{L} \subset \mathbb{R}^{2n}$ of dimension $n$ - which is called a Lagrangian subspace - the question whether $\mathcal{L}$ can be generated as the Krylov space of some skew-Hamiltonian matrix is considered. The affine variety $\mathbb{HK}$ of all skew-Hamiltonian matrices $H \in \mathbb{R}^{2n \times 2n}$ that generate $\mathcal{L}$ as a Krylov space is analyzed. Existence and uniqueness results are proven, the dimension of $\mathbb{HK}$ is found and skew-Hamiltonian matrices with minimal $2$-norm and Frobenius norm in $\mathbb{HK}$ are identified. In addition, a simple algorithm is presented to find a basis of $\mathbb{HK}$.