On skew-Hamiltonian Matrices and their Krylov-Lagrangian Subspaces

TL;DR

This paper investigates the conditions under which a Lagrangian subspace can be generated as a Krylov space by skew-Hamiltonian matrices, analyzing the structure, existence, uniqueness, and minimal norm solutions within this class.

Contribution

It characterizes the affine variety of skew-Hamiltonian matrices generating a given Lagrangian subspace as a Krylov space, providing existence, uniqueness, and minimal norm results, along with an algorithm.

Findings

Dimension of the variety of such matrices is determined.

Existence and uniqueness conditions are established.

Algorithms for basis computation are developed.

Abstract

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 $\mathbb{HK}$.