# How can I prove that $\mathbb{S}_{+}^n$ is a closed and convex set?

$$\mathbb{S}_{+}^n$$ is the set of positive semidefinite (and symmetric) real matrices of size $$n\times n$$. I have to prove that this set is a closed convex cone. How can I do?

• Do you know the definitions of closed, convex, and cone? Can you show any of the three properties? – Mees de Vries Jan 16 at 13:49
• It is not closed. ${ 1 \over n} I \to 0$. The other properties are just a matter of grinding through the definitions. – copper.hat Jan 16 at 15:05
• I expect positive means positive definite as in $Ax\cdot x\geq0$ for all $x$. – SmileyCraft Jan 16 at 15:18
• @SmileyCraft that would be positive semidefinite – LinAlg Jan 16 at 15:31

Note that $$A \ge 0$$ iff $$x^TAx \ge 0$$ for all $$x$$ iff $$A \in \cap_x \{ B | x^T B x \ge 0\}$$.
Note that for any $$x$$ that $$\{ B | x^T B x \ge 0\}$$ is a closed half space (hence convex). Since $$0 \in \{ B | x^T B x \ge 0\}$$ we see that it is a cone as well.