Okay, a bit of linear algebra: Linear operators M have a set of eigenvectors vi such that ##Mv_i = \lambda_i v_i## for some constant (the eigenvalue) ##\lambda_i##. As an example, an operator "mirror the z-direction" in 3D space would have the x- and y-direction as eigenvectors with eigenvalue 1 and the z-direction as eigenvector with eigenvalue -1. "Mirror across the plane x=z" would have the y-axis and a vector (1,0,1) as eigenvector with eigenvalue 1 and the vector (1,0,-1) as eigenvector with eigenvalue -1.
In general different operators have different eigenvectors - if you apply an operator on an eigenvector of a different operator you get something, but in general not a multiple of this eigenvector. There is an exception: If two operators commute, then you can find a set of eigenvectors for both operators together. For operators M,N "commute" means ##MNv=NMv## for all v. For people, "commute" means something else...