Why does the set of positive definite matrices define a half-space?

Question

A half-space is a set of the form $\{x \mid a^Tx \leq b\}$. Also it is stated that the set $$\{X \in S^n \mid z^T X z \geq 0 \}$$ where $S^n$ denote the set of symmetric $n\times n$ matrices, is a half space$^1$,

1. Can we expand it algebraically to prove it?
2. Is is possible to make a graphical intuitive example, e.g in matlab, as the one that can easily be made for $\{x \mid a^Tx \leq b\}$?

Thanks!!

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^{$^{1}$Convex Optimization by Boyd & Vandenberghe, pp.36}

Answer 1

$z^\top X z = (z \otimes z)^\top vec(X)$.

Now, $(z \otimes z)$ is a vector.

Answer 2

Correction of the answer above, it is not (), it is infinite intersection of infinite half-spaces in n D symmetric matrix dimension. If x is vector the it belongs to Rn not Sn