Quadratic bosonic Hamiltonian ($H=a^{†}Aa+\frac{1}{2}(a^{T}Ba+h.c.)$)

A quadratic bosonic Hamiltonian contains terms of degree two in bosonic creation operators and bosonic annihilation operators, with an optional constant. The number-conserving matrix $A$ is Hermitian and the pair matrix $B$ may be taken symmetric. Positive stable forms can be diagonalized by a canonical Bogoliubov transformation; the strict stability condition matters at zero modes.