closed, convex cone C $\in \mathbb{R}^n$ whose linear hull is the entire $\mathbb{R}^2$

Question

I am trying to construct the following example:

let $A \in \mathbb{R}^{2x2}$ be symmetric matrix and C $\in \mathbb{R^n}$ a closed, convex cone whose linear hull is the entire $\mathbb{R}^2$ and

$x^TAx >0 \ \forall x \in C \backslash \{0\} $,

but $A$ not positive semi-definite.

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I constructed $A$ as follows:

$ A = \left( \begin{array}{rr} 1 & 0 \\ 0 & 1\\ \end{array} \right)$, such that $x^T A x = x_1^2 + x_2^2 > 0 $

I am completely stuck on constructing a cone that will fullfill above conditions. Any input is highly appretiated.

Answer 1

Choose $\alpha \in (0,1)$ and let $C = \{ (x \mid \alpha x_1 \ge |x_2| \}$. Let $A = \operatorname{diag} (1,-1)$. Then if $x \in C\setminus \{0\}$ we have $x^TAx =x_1^2-x_2^2 \ge (1-\alpha^2) x_1^2+ \alpha x_1^2-x_2^2 \ge (1-\alpha^2) x_1^2 > 0$. It should be clear that since $(1,\pm \alpha) \in C$ that the span of $C$ is the entire plane.