  A system of linearly coupled quantum harmonic oscillators can be diagonalized
when the system is dynamically stable using a Bogoliubov canonical
transformation. However, this is just a particular case of more general
canonical transformations that can be performed even when the system is
dynamically unstable. Specific canonical transformations can transform a
quadratic Hamiltonian into a normal form, which greatly helps to elucidate the
underlying physics of the system. Here, we provide a self-contained review of
the normal form of a quadratic Hamiltonian as well as step-by-step instructions
to construct the corresponding canonical transformation for the most general
case. Among other examples, we show how the standard two-mode Hamiltonian with
a quadratic position coupling presents, in the stability diagram, all the
possible normal forms corresponding to different types of dynamical
instabilities.
