Let

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 .

And .

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 .