[brian_cloutier]

If anyone wants to read this in book form, Linear Algebra Done Right includes it as an exercise at the end of a very short and readable chapter 5.C on eigenspaces and diagonal matricies.

The treatment in thirty-three miniatures describes the steps you take but doesn't mention what you are actually doing (finding eigenvalues) or leave you with any intuition for why this was a natural thing to have tried

[abhi9u]

Yes, that book (thirty three miniatures) has great content, but a hard read. Basically someone like me needs to go back, read other sources, and spend time on paper to get it.