On skew-Hamiltonian Matrices and their Krylov-Lagrangian Subspaces
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It is a well-known fact that the Krylov space K_j(H,x) generated by a skew-Hamiltonian matrix H ∈R^2n × 2n and some x ∈R^2n is isotropic for any j ∈N. For any given isotropic subspace L⊂R^2n 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 ∈R^2n × 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 and Frobenius norm in HK are identified. In addition, a simple algorithm is presented to find a basis of HK.
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