Poisson bracket in complex canonical coordinates

For $z_j=q_j+i{p}_j$ and ${\bar{z}}_j=q_j-i{p}_j$,This follows from ${\partial }_{q_j}={\partial }_{z_j}+{\partial }_{{\bar{z}}_j}$ and ${\partial }_{p_j}=i{\partial }_{z_j}-i{\partial }_{{\bar{z}}_j}$.

For an anti-Hermitian matrix $A$, the real quadratic HamiltoniansatisfiesThus $A\mapsto -I_A$ realizes the matrix commutator as a Poisson bracket; it is the quadratic moment-map construction for the unitary action.