On skew-Hamiltonian matrices and their Krylov-Lagrangian subspaces

Abstract

It is a well-known fact that the Krylov space Kj(H,x) generated by a skew-Hamiltonian matrix H∈R2n×2n and some x∈R2n is isotropic for any j∈N. For any given isotropic subspace L⊂R2n of dimension n—which is called a Lagrangian subspace—the question whether L can be generated as the Krylov space of some skew-Hamiltonian matrix is considered. The affine variety HK of all skew-Hamiltonian matrices H∈R2n×2n that generate L as a Krylov space is analyzed. Existence and uniqueness results are proven, the dimension of HK is found and skew-Hamiltonian matrices with minimal 2-norm, minimal Frobenius norm and prescribed eigenvalues in HK are identified. Some applications of the presented results are given.