Consider the Hamiltonian system

$$\mathbf{p}^{\prime}=-\frac{\partial H}{\partial \mathbf{q}}, \quad \mathbf{q}^{\prime}=\frac{\partial H}{\partial \mathbf{p}}$$

where $H=H(\mathbf{p}, \mathbf{q})$.

When is the transformation $\mathbf{P}=\mathbf{P}(\mathbf{p}, \mathbf{q}), \mathbf{Q}=\mathbf{Q}(\mathbf{p}, \mathbf{q})$ canonical?

Prove that, if the transformation is canonical, then the equations in the new variables $(\mathbf{P}, \mathbf{Q})$ are also Hamiltonian, with the same Hamiltonian function $H$.

Let $\mathbf{P}=C^{-1} \mathbf{p}+B q, \mathbf{Q}=C \mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation to be canonical.