Set of symmetric positive semidefinite matrices is a full dimensional convex cone.

Solution 1:

For closed, note that the functions $f_1:\mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}$ given by $f_1(A) = A-A^T$, and $f_2: \mathbb{R}^{n\times n} \to \mathbb{R}$ given by $f_2(A) = \min_{||x||=1} \langle x, A x \rangle$ are continuous, so the sets $f_1^{-1} \{ 0 \}$ and $f_2^{-1} [0,\infty)$ are closed.

For convex, it is straightforward to check that if $\lambda \in [0,1]$, and $A,B \in S_n^+$, then $\lambda A + (1-\lambda)B \in S_n^+$ (just use the definition of positive semi-definiteness).

For cone, again it is straightforward to check that if $t\geq 0$ (0r whatever your definition of cone allows), then when $A \in S_n^+$, then $t A \in S_n^+$ (again, just use the definition).