Why the final equality in this verification of positive semi-definiteness? In showing that a particular Hessian is positive semidefinite the author writes:
$$
\frac{2}{y^3} \left[ \begin{array}{cc} y^2 & -xy \\ -xy & x^2 \end{array} \right] = \frac{2}{y^3} \left[ \begin{array}{c} y & -x \\ \end{array} \right]^T \left[ \begin{array}{c} y & -x \\ \end{array} \right]$$
I can see from the left hand side that the matrix is positive semidefinite but I'm not sure what the factorization on the right is getting at or making reference too in terms of positive semidefiniteness of a matrix..

I presume $y > 0$, else it's not true.
Any matrix of the form $A^T A$ where $A$ is a real $m \times n$ matrix (or in this case a vector, which can be considered as an $m \times 1$ matrix) is positive semidefinite.  This is because for any real vector $v$, 
$v^T A^T A v = (A v)^T (A v) \ge 0$.   
