Question

The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 matrices.

Does this property generalize to solutions of linear systems of semidefinite matrices?

Let me be precise. Fix $k$ symmetric $n\times n$ matrices $A_1,\ldots, A_k$. Consider the system of linear equations $\langle A_1,X\rangle=\cdots = \langle A_k, X\rangle = 0$, which you want to solve for a symmetric semidefinite $n\times n$ matrix $X\succeq 0$. Here, the inner product of two matrices is $\langle(a_{ij}),(b_{ij})\rangle=\sum_{i,j = 1}^n a_{ij}b_{ij}$.

The set of solutions $X$ forms a closed convex subcone $C$ of the cone of semidefinite $n\times n$ matrices. Is $C$ the convex hull of its rank 1 matrices? Namely, is every solution a convex combination of rank 1 solutions?

Answer 1

No. Try $n=k=2$ with $A_1 = \pmatrix{1 & 0\cr 0 & -1\cr}$ and $A_2 = \pmatrix{1 & 1\cr 1 & -1\cr}$. The only symmetric matrices $X$ with $(A_1,X) = (A_2,X) = 0$ are multiples of $I$, so there are no rank 1 solutions.

Answer 2

The answer seems obviously "no", since a system of equations need not have any rank one solutions (for example, the solution set can be the line $x I_n,$ where $I_n$ is the identity matrix.)