Consider the following canonical transformation: η η = [ q 1 ⋮ ⋮ q N p 1 ⋮ ⋮ p N ] → → ε ε = [ Q 1 ⋮ ⋮ Q N P 1 ⋮ ⋮ P N ] Defining M := ∂ ∂ ( Q , P ) ∂ ∂ ( q , p ) , the Poisson matrix is defined as P ( ε ε ) = M J M T , where J is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that: P i j ( ε ε ) = [ M J M T ] i j = ∑ ∑ k = 1 N ( ∂ ∂ ε ε i ∂ ∂ η η k ∂ ∂ ε ε j ∂ ∂ η η N + k − − ∂ ∂ ε ε i ∂ ∂ η η N + k ∂ ∂ ε ε j ∂ ∂ η η k ) = ∑ ∑ k = 1 N ( ∂ ∂ ε ε i ∂ ∂ q k ∂ ∂ ε ε j ∂ ∂ p k − − ∂ ∂ ε ε i ∂ ∂ p k ∂ ∂ ε ε j ∂ ∂ q k ) = { ε ε i , ε ε j } η η .