  The classical spectral theorem completely describes self-adjoint operators on
finite dimensional inner product vector spaces as linear combinations of
orthogonal projections onto pairwise orthogonal subspaces. We prove a similar
theorem for self-adjoint operators on finite dimensional symplectic vector
spaces over perfect fields. We show that these operators decompose according to
a polarization, i.e. as the product of an operator on a lagrangian subspace and
its dual on a complementary lagrangian. Moreover, if all the eigenvalues of the
operator are in the base field, then there exist a Darboux basis such that the
matrix representation of the operator is two-by-two blocks block-diagonal,
where the first block is in Jordan normal form and the second block is the
transpose of the first one.
