Given Equation of Inner product , how to determine that its positive for all $x,y\in \mathbb{R}$

Question

I'm confused on how to configure if an equation is positive for all $x,y$.

for example the inner product : $<(x,y),(x,y)> =2x^2+y^2-2xy$
How can I know if this equation is positive or it might be negative for some $x,y$ ?

Thank you in advance.

Answer 1

Hint:

Observe that $x^2\ge 0$ and $ x^2+y^2-2xy=(x-y)^2\ge 0 $.

In general, what you need is this: https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia