Can $z + \frac{1}{z}$ or $z + \frac{2}{z}$ be considered as quadratic forms?

Question

Can $$z + \frac{1}{z}$$ or $$z + \frac{2}{z}$$ be considered as quadratic forms?

Answer 1

Let $x=\sqrt z$ and $y=\dfrac1{\sqrt z}$.

Then, $z+\dfrac1z=1x^2+0xy+1y^2$ and $z+\dfrac2z=1x^2+0xy+2y^2$.

However, this only works when $z\ge0$.