  Ground states of quadratic Hamiltonians for fermionic systems can be
characterized in terms of orthogonal complex structures. The standard way in
which such Hamiltonians are diagonalized makes use of a certain "doubling" of
the Hilbert space. In this work we show that this redundancy in the Hilbert
space can be completely lifted if the relevant orthogonal structure is taken
into account. Such an approach allows for a treatment of Majorana fermions
which is both physically and mathematically transparent. Furthermore, an
explicit connection between orthogonal complex structures and the topological
$\mathbb Z_2$-invariant is given.
