What Matrices Commute with a Diagonal Matrix with Distinct Eigenvalues?

[jgens]

As part of a larger problem involving classifying intertwining operators of two group representations, I came across the following question: If $X$ is an $n \times n$ diagonal matrix with $n$ distinct non-zero eigenvalues, then exactly which $n \times n$ matrices $A$ satisfy the following equality $AXA^{-1} = X$? Does anyone know the answer to this question?

Edit: Nevermind. I found a better way of doing the problem that avoids this sort of argument.

 

[HallsofIvy]

Those whose eigenvalues are the numbers on the diagonal of the original matrix.

 

[Vargo]

Is that true? I believe it is the set of operators with the same invariant subspaces. The eigenvalues don't have to be the same, they just have to be simultaneously diagonalizable.