1. The problem statement, all variables and given/known data A and B are commuting diagonalizable matrices. Prove that they are simultaneously diagonalizable. 2. Relevant equations AB = BA 3. The attempt at a solution I have what looks like a proof, but I'm not very happy with it. Is there anything wrong here? AB = BA B = ABA-1 Every matrix has exactly one jordan form. All diagonal matrices are jordan forms. So there exists a unique marix C such that CBC-1 = J(C) where J(C) is C's Jordan form. J(C) = CBC-1 = CABA-1C-1 = (CA)B(CA)-1 As C is unique, C = CA, so A = I And CIC-1 = I, which is diagonal. So A and B are simultaneously diagonalisable.