I can do i) but just a bit stuck on ii).

my working for i) is: E^2-E=0 so that means $\displaystyle \lambda^2-\lambda$ is an eigenvalue of E^2-E=0, since the zero vector only has an eigenvalue of 0, then $\displaystyle \lambda^2-\lambda=0$ and so \lambda = 0 or 1

cheers

[hr]

I sorta have an idea but dono how to finish this Q

If we let x be a eigenvector of $\displaystyle B^TB$ corresponding to $\displaystyle \lambda$, then we have $\displaystyle B^TBx = \lambda x$

Now just playing around: $\displaystyle B^TBx.x = x. ((B^TB)^Tx)=x.(B^TB)x= x.(\lambda x)$

but dono how to get to the result $\displaystyle \lambda \ge 0$