1. The problem statement, all variables and given/known data A and B are commuting diagonalizable linear operators. prove that they are simultaneously diagonalizable. 2. Relevant equations AB = BA 3. The attempt at a solution We deal with the problem in the Jordan basis of A, where A is diagonal, as Jordan forms are unique. Then by rearranging the basis vectors, we can treat A as a block diagonal matrix, where the blocks are of the form λiI. I aim to prove that, if A is diagonal, and commutes with B, then B must also be diagonal, so they have the same Jordan basis. I can prove that B must also be a block diagonal matrix, with the dimensions of the blocks mirroring those of A. This is because if a nonzero entry exists outside of and of B's blocks, the corresponding entries in AB and BA would be this entry multiplied by different eigenvectors. So the multiplication would not be commutative. But from here I don't know what to do next. Is there some restriction that a diagonalizable matrix may not be put in block diagonal form where the blocks are not diagonalizable themselves?