Suppose $A$ and $B$ are both diagonalizable $n \times n$ matrices where $A$ has eigenvector basis $\{\mathbf{v}_1, \ldots, \mathbf{v}_n\}$ and $B$ has completely different eigenvector basis $\{\mathbf{w}_1, \ldots, \mathbf{w}_n\}$ with no vectors in common. Which statement must be true?