The structure of alternating-Hamiltonian matrices

Generalizing results proved recently for the real and complex case, we show over all fields that every alternating-Hamiltonian matrix is similar to a block-diagonal matrix of the form A00At, and that any two similar ones are similar by a symplectic transformation. Furthermore, every one is a square of a Hamiltonian matrix. The proofs use a structural idea drawn from the study of pairs of alternating forms. Counterexamples show that the definitions must be carefully chosen to work in characteristic 2.