Then we solve the equation for each , we then arrive:
Therefore and .
Rayleigh quotient? See page 2 of this document for the proof of maximum and minimum. So must be an eigenvector of . We also have:
So if is an eigenvector of then must also be the same eigenvector of . If we keep do this recursion, we will finally arrive the conclusion that must also be the same eigenvector of .
, attains the maximum value at and attains the minimum value at .