Let then .
Let . If this basis diagnolizes then it means is a diagnol matrix.
Let be the transition matrix from to then it means where is coordinate vector with respect to and is coordinate vector with respect to .
Now find eigenvalues of .
Then it would mean would diagnolize .
Now this means
If you evaluate this equality at you get
Thus, we get where is first colomn of . It follows that and that gives you what is. Now do the same procedure for .