Prove that if two matrices $A$ and $B$ commute, $A$ is diagonal, and all its diagonal values are distinct, then $B$ is also diagonal.

Question

How could I show this?

I know since they commute, $A$ and $B$ have the same set of eigenvectors and they can be diagonalized simultaneously. But I have trouble showing the above statement. Any help would be appreciated!

Answer 1

Suppose that $A={\rm diag}(a_1,\ldots ,a_n)$, and $B=(b_{ij})$. Then $AB=BA$ gives equations $(a_i-a_j)b_{kl}=0$ for $k\neq l$. This implies $b_{kl}=0$ for all $k\neq l$.