Denote by and respectively the set of complex matrices and the identity matrix of order . Set . A matrix is Hamiltonian (resp. skew-Hamiltonian) if with blocks in where , , (resp. , , ); one has (resp. ). A matrix is symplectic if .
The author proves that every skew-Hamiltonian matrix can be brought into skew-Hamiltonian Jordan form where is in complex Jordan form, by a symplectic similarity transformation. This is the complex analog of a similar result of H. Fassbender, D. S. Mackey, N. Mackey, and H. Xu in the case of real matrices [Linear Algebra Appl. 287, No. 1-3, 125-159 (1999; Zbl 0940.15017)]. The author proves also that every skew-Hamiltonian matrix has a Hamiltonian square root, that two similar skew-Hamiltonian matrices , are symplectic similar, i.e. , and that every nonsingular matrix is representable in the forms and ; here , are symplectic and , are skew-Hamiltonian.