The master equation can be simplified so that the terms with ℓ = k do not appear in the summation. This allows calculations even if the main diagonal of A is not defined or has been assigned an arbitrary value. d P k d t = ∑ ∑ ℓ ℓ ( A k ℓ ℓ P ℓ ℓ ) = ∑ ∑ ℓ ℓ ≠ ≠ k ( A k ℓ ℓ P ℓ ℓ ) + A k k P k = ∑ ∑ ℓ ℓ ≠ ≠ k ( A k ℓ ℓ P ℓ ℓ − − A ℓ ℓ k P k ) .