I just finished my Linear Algebra exam and this ONE question gave me trouble...

Can someone PLEASE let me know how I should have proceeded with this question if I didn't already go about it the right way?

Question: Show that the eigenvalues for (A^T)(A) are positive where A is a n x n matrix.

What I did: I let A be a 2 x 2 matrix with a, b, c, d comprising it, and I transposed them and multiplied them together to get (A^T)(A) and then I partially computed the characteristic polynomial but I was left with a bunch of terms that led me nowhere unfortunately - just a bunch of collected like terms with no way of verifying that the eigenvalues are always positive.

Your approach isn't a particularly useful one. Even if you could express the eigenvalues in terms of a, b, c, and d you would at best show it was true for the 2x2 case.

I'd use the property of the transpose

y•(Ax) = (A^Ty)•x

for any x, y in IRⁿ (we are told that A is n x n). So suppose that (λ, x) is an eigenvalue eigenvector pair of A^TA. Then

x•(A^TAx) = λ ||x||²

And using the above and the fact that (A^T)^T = A you get

λ ||x||² = x•(A^TAx) = (Ax)•(Ax) = ||Ax||².

Since ||x||² can not be zero (it's an eigenvector)

λ = ||Ax||²/||x||² ≥ 0.

Incidentally, you can't say that they are "positive", you can only say that they are non-negative. They can be zero.