Now for a subtlety. The implied summation on the right hand side of A ′ ν ν = Λ Λ ν ν μ μ A μ μ = ( Λ Λ − − 1 ) μ μ ν ν A μ μ is running over a row index of the matrix representing Λ. Thus, in terms of matrices, this transformation should be thought of as the inverse transpose of Λ acting on the column vector A μ. That is, in pure matrix notation, A ′ = ( Λ Λ − − 1 ) T A .