# Todai Entrance Exam: Math 2017 – Problem 1

**(1)**

Therefore:

**(2)**

Then we solve the equation for each , we then arrive:

**(3)**

**(4)**

As:

Therefore:

So:

**(5)**

Notice:

Therefore and .

**(6)**

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 .