Comments for 30

23 24 and 26 were good

A lot of people messed up on 25. My solution is as follows:

Assume B is positive definite, B^2=A. Then one can check easily that any
eigenvector of B is also an eigenvector of A, with the A-eigenvalue being the
square of the corresponding B-eigenvalue. Since the eigenvectors of both are
lin. ind. (ie distinct) by the PAT, we know that B and A have the same number
of eigenvectors, and hence the same eigenvectors. It follows that B must be of
the form shown at the top of page 24, and so B is defined uniquely in terms of
A. QED.

Some people just did the hint without showing uniqueness, and there were a few
other common mess-ups. I got the feeling that most people (ie almost everybody)
didnt really know how to do it so they just reiterated the hint and hoped they
would get some points.