# When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices?

Noferini, Vanni (2014) When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices? [MIMS Preprint]

The mapping $\Phi_n(A,B)=AB-BA$, where the matrices $A,B \in \mathbb{C}^{2n \times 2n}$ are skew-Hamiltonian with respect to transposition, is studied. Let $\mathcal{C}_n$ be the range of $\Phi_n$: we give an implicit characterization of $\mathcal{C}_n$, obtaining results that find an application in algebraic geometry. Namely, they are used in [R. Abuaf and A. Boralevi, Orthogonal bundles and skew-Hamiltonian matrices, In Preparation] to study orthogonal vector bundles. We also give alternative and more explicit characterizations of $\mathcal{C}_n$ for $n \leq 3$. Moreover, we prove that for $n \geq 4$ the complement of $\mathcal{C}_n$ is nowhere dense in the set of $2n$-dimensional Hamiltonian matrices, denoted by $\mathcal{H}_n$, implying that almost all matrices in $\mathcal{H}_n$ are in $\mathcal{C}_n$ for $n \geq 4$. Finally, we show that $\Phi_n$ is never surjective as a mapping from $\mathcal{W}_n \times \mathcal{W}_n$ to $\mathcal{H}_n$, where $\mathcal{W}_n$ is the set of $2n$-dimensional skew-Hamiltonian matrices. Along the way, we discuss the connections of this problem with several existing results in matrix theory.