  In this work some results on the structure-preserving diagonalization of
selfadjoint and skewadjoint matrices in indefinite inner product spaces are
presented. In particular, necessary and sufficient conditions on the symplectic
diagonalizability of (skew)-Hamiltonian matrices and the perplectic
diagonalizability of per(skew)-Hermitian matrices are provided. Assuming the
structured matrix at hand is additionally normal, it is shown that any
symplectic or perplectic diagonalization can always be constructed to be
unitary. As a consequence of this fact, the existence of a unitary,
structure-preserving diagonalization is equivalent to the existence of a
specially structured additive decomposition of such matrices. The implications
of this decomposition are illustrated by several examples.
