Kreyszig is one of the most readable math books out there, you can't go wrong with it. You'll definitely want to pick it up either way if you are continuing in any field where you may need functional analysis.
A matrix is PSD if and only if it's principal minors are nonnegative (see http://en.wikipedia.org/wiki/Positive-definite_matrix#Characterizations").
A 2x2 matrix has three principal minors - the diagonal elements, and the determinant. So x,z >= 0, and xz - y^2 >=0.
I'm sure there is a way to...