The final equality arises from the fact that ∑ ∑ ℓ ℓ , k ( A ℓ ℓ k P k ) = d d t ∑ ∑ ℓ ℓ ( P ℓ ℓ ) = 0 because the summation over the probabilities P ℓ ℓ yields one, a constant function. Since this has to hold for any probability P → → (and in particular for any probability of the form P ℓ ℓ = δ δ ℓ ℓ k for some k) we get ∑ ∑ ℓ ℓ ( A ℓ ℓ k ) = 0 ∀ ∀ k . Using this we can write the diagonal elements as A k k = − − ∑ ∑ ℓ ℓ ≠ ≠ k ( A ℓ ℓ k ) ⇒ ⇒ A k k P k = − − ∑ ∑ ℓ ℓ ≠ ≠ k ( A ℓ ℓ k P k ) .