First, note that I have made some corrections to the question. The most important one is that some of the UU*s should be U*U.

Start by doing what the Hint suggests, and multiply the matrices to get the product

.

Now comes the clever part. The Hint suggests that you should write this in the form . But (because ). So we get

In the equation , you can see that if is positive definite then the right side of the equation will be positive for all v and w. Therefore the left side is positive, which means that the matrix is positive definite. On the other hand, if the matrix is positive definite then the left side of the equation is positive for all v and w, and in particular when . That implies that is positive for all v, which tells us that is positive definite.

Thus condition (a) is equivalent to condition (b). A similar calculation shows that (a) is equivalent to (c).