The application of U to the whole matrix H yields then H D = U − − 1 H U = D 0 + V ′ with a transformed perturbation V ′ = U − − 1 V U , which remains off-diagonal. In this new form, the second step to compute S becomes very simple, since we obtain an explicit expression, in components: S i j = V i j ′ d i − − d j where d i denotes the i th element on the diagonal of D 0 . The reason for this comes from the observation that, for any matrix A = ( A i j ) , and diagonal matrix D = diag ⁡ ⁡ ( d i ) , we have the relation [ D , A ] i j = ( d i − − d j ) A i j . Since the generator for H D is defined by [ D 0 , S ] = V ′ , the above formula follows immediately. As expected, the associated operator W = e S is unitary (it satisfies W † † = W − − 1 ) because the denominator of S changes sign when transposed, and V is Hermitian.