For a set of $m$ positive semi-definite $dtimes d$ matrices $Q_j$ I have the following system of equations over column-vector $vec{x}$:

$$

q_j = vec{x}^T Q_j vec{x}, quad j=1,dots,m

$$

with some known non-negative $q_j$.

- Are there effective numerical solvers for this type of systems?
- For a given solution $vec{y}$ there also exists the solution $-vec{y}$. Is there a way to prove that there are no other solutions exists for the specific problem?