On the diagonalizability of a matrix by a symplectic equivalence, similarity or congruence transformation

A symplectic matrix S∈C2n×2nS∈C2n×2n satisfies S=J−1STJS=J−1STJ for J=[0−InIn0]∈R2n×2n. We will consider symplectic equivalence, similarity and congruence transformations and answer the question under which conditions a 2n×2n2n×2n matrix is diagonalizable under one of these transformations. In particular, we will give symplectic analogues of the singular value decomposition and the Takagi factorization.