A Comment about the Lambdas

 

• It is easy to build matrices $\Lambda^l{}_{l'}$ and $\Lambda^{l'}{}_l$ from eigenvectors.
• Since $\boldsymbol{e}_{l'} = \boldsymbol{e}_{l}\Lambda^l{}_{l'}$ we see that if we were to identify e_l with vertical columns of zeros with ones corresponding to l, then $\Lambda^l{}_{l'}$ would be made of coefficients of e_l' in that basis, that is the first column of $\Lambda^l{}_{l'}$ would be e_1', the second column of $\Lambda^l{}_{l'}$ would be e_2' and so on. If F is normal, then $\boldsymbol{\Lambda} \in \hbox{SO}(n)$ and $\Lambda^{l'}{}_l$ is simply a transpose of $\Lambda^l{}_{l'}$, that is, its first row is a transposed $\boldsymbol{e}_{1'}{}^T = \boldsymbol{\omega}^{1'}$, its second row is a transposed $\boldsymbol{e}_{2'}{}^T = \boldsymbol{\omega}^{2'}$, and so on.