What happens if two operators commute?

[Master J]

I am trying to understand the idea of measurements on a system. Forgive me if any of my interpretations are incorrect...I'm hoping things can be cleared up.


A measurement is taken on a system, represented by an operator, and this measurement changes the state of the system into a state which corresponds to an eigenfunction of the operator.

If a different measurement is then taken, and this operator commutes with the previous one, does that mean that both operators have the same eigenfunctions? I arrived at this because, if they commute, the order they are applied won't matter, so the state of the system should be the same?
Of course, if they don't commute, one measurement changes the state of the system so that the system is now different when one takes the second measurement, affecting its result.

 

[VanOosten]

yes if two operators commute they share the same eigenfunctions and can be measured at the same time with 100% certainty in both

 

[arkajad]

If two operators commute, then one can choose common eigenfunctions, but that does not mean that every eigenfunctions of A is an eigenfunction of B. Take for instance A=Identity operator.