How to prove a set of positive semi definite matrices forms a convex set?

Question

Let $C$ be the set of positive semi-definite matrices, how can I prove it is a convex set?

Answer 1

A matrix is positive semi-definite (notation $A \succeq 0$) iff $x^{T} A x \ge 0$ for all $x\in \mathbb{C}^n$.

If $A\succeq 0, B \succeq 0$, then if $\lambda \in [0,1]$ we have $ x^{T}( \lambda A + (1-\lambda)B )x = \lambda x^{T} A x + (1-\lambda) x^{T} B x \ge 0$.Hence $\lambda A + (1-\lambda)B \succeq 0$.

Alternatively, you could fix $x$, note that $A \mapsto x^T A x $ is a linear functional on the space of matrices, and hence the set $H_x = \{ A | x^T A x \ge 0 \}$ is convex (a halfspace when $x \neq 0$).

Then we have $\{ A | A \succeq 0 \} = \cap_{x} H_x$, and since the intersection of convex sets is again convex, we see that the set of positive semi-definite matrices is convex.

Essentially the same reasoning applies to positive definite, negative definite and negative semi-definite.