1. The problem statement, all variables and given/known data In a given basis, the eigenvectors A and B are represented by the following matrices: A = [ 1 0 0 ] B = [ 2 0 0 ] [ 0 -1 0] [ 0 0 -2i ] [ 0 0 -1] [ 0 2i 0 ] What are A and B's eigenvalues? Determine [A, B]. Obtain a set of eigenvectors common to A and B. Do they form a complete basis? 2. Relevant equations (A - λI)x = 0 [A, B] = AB - BA 3. The attempt at a solution Okay, so, I calculated the eigenvalues and the commutator quite easily. For A, I got eigenvalues 1 and -1, with -1 having degeneracy 2. For B, I got eigenvalues 2 and -2, with 2 having degeneracy 2. The commutator was 0, so they commutate. Now, as far as common eigenvectors go - I could only find one. [1 0 0] transposed. Is this due to the eigenvalues having degeneracy? Does the fact that two observables commuting implies that they have a common complete basis of eigenvectors only hold up if they don't come from degenerate eigenvalues? Thank you for your help -- the material given to me was not very clear regarding this particular case.