Q:

Prove two convex sets are equal

Prove that following two sets are equal:
  $$
\operatorname{conv}\left\{\, xx^T \,\middle|\, x\in\Bbb R^n, \|x\|=1 \,\right\} = \left\{ A \in S_n^+ \,\middle|\, \operatorname{Tr}(A)=1 \,\right\},
$$
  where $S_n^+$ is the set of all positive-semidefinite matrices.

I already proved that second one contains the first (it is quite easy), but I have no idea how to prove that every positive-semidefinite matrix is the weighted sum of outer products.

A:

Every symmetric matrix can be decomposed as follows:
$$
{\displaystyle A=Q\Lambda Q^{\textsf {T}}},
$$
where $\Lambda$ is diagonal
We can construct a diagonal matrix from outer products easily, taking vectors with only one non-zero entry of 1 and coefs as diagonal entries of the matrix. Then we just insert $Q$ and $Q^{\textsf {T}}$ into the sum and get the weighted sum of outer products.

