1. The problem statement, all variables and given/known data Given a Hermitian operator A = [itex]\sum \left|a\right\rangle a \left\langle a\right|[/itex] and B any operator (in general, not Hermitian) such that [itex]\left[A,B\right][/itex] = [itex]\lambda[/itex]B show that B[itex]\left|a\right\rangle[/itex] = const. [itex]\left|a\right\rangle[/itex] 2. Relevant equations Basically those listed above plus possibly the Hermitian condition and eigenvalue definition which I will not list since they are well known. 3. The attempt at a solution I have tried expanding it out in terms of the commutator, but this seems like the wrong approach. I am not sure that there is a way to calculate it directly. I do not think the proof is very involved but I am approaching it the wrong way and cant seem to get anywhere. This is listed as a fundamental property of a Hermitian operator in a complex vector space in my text, though it does not prove it and other references seem unconcerned. I am also not very comfortable using this kind of spectral decomposition, which is a bit of a problem.