Let $C = \mathbb{S_{n}^{+}}$ be the set of symmetric, positive semi-defnite matrices (which is a closed,convex set).
Given, $$Z = \begin{pmatrix} 3 & 1 & -1\\ 1 & 1 & 1\\ -1 & 1 & 9\\ \end{pmatrix}$$
Find the normal cone, $N_{C}(Z).$
By definition, $$N_{C}(Z):= \{A \in M_{n \times n}(\mathbb{R}): A^{T}(B-Z) \leq 0, \forall B \in C\}$$
I am not able to find such $A$ and $B$. Is there a more efficient way to tackle this problem?