I know how to diagonalize a given quadratic form using the Gaussian method. Though, I once read somewhere that there's another method which uses an augmented matrix, but I didn't go into details.

Say: $q: \mathbb R^2 \to \mathbb R$, $q(x,y) = xy - y^2$.
How would I use that method to diagonalize $q$?
$$xy - y^2 = \left( \frac{x}{2} \right)^2 - \left( \frac{x}{2} - y \right)^2$$