Exercise 2

Given that \(\mathbf M\) and \(\mathbf L\) are Hermitian,

\[ \mathbf M^\dagger = \mathbf M \] \[ \mathbf L^\dagger = \mathbf L \]

then \(\mathrm i [\mathbf M, \mathbf L]\) is also Hermitian:

\[ \begin{flalign*} (\mathrm i [\mathbf M, \mathbf L])^\dagger &= (\mathrm i(\mathbf M \mathbf L - \mathbf L \mathbf M))^\dagger \\ &= \mathrm i^*((\mathbf M \mathbf L - \mathbf L \mathbf M))^\dagger \\ &= -\mathrm i(\mathbf L^\dagger \mathbf M^\dagger - \mathbf M^\dagger \mathbf L^\dagger)\\ &= -\mathrm i(\mathbf L \mathbf M - \mathbf M \mathbf L)\\ &= \mathrm i(\mathbf M \mathbf L - \mathbf L \mathbf M) \\ &= \mathrm i[\mathbf M, \mathbf L] \\ \end{flalign*} \]