What does it mean for a polynomial to be positive semi-definite? I'm reading proofs about Cauchy-Schwarz inequality.
Some of them use the argument of a polynomial being positive semi-definite as the starting argument.
That is, they argue that
$$\forall x,y \in\mathbb{R^n}, a\in\mathbb{R}\\\langle {ax+y,ax+y} \rangle$$
Is a positive semi-definite polynomial (when expanded). How is this seen?
There's one proof here if you can see it (page 2):

It means that the expression $$\langle ax+y, ax+y\rangle$$
is always greater than or equal to zero.
This is true because the inequality $\langle x,x\rangle \geq 0$ is true for any element $x$ in your vector space.
