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 eigenspac

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