Consider a vector space $V \in \mathbb{R}^n$. Given a set of $\frac{1}{2}n(n-1)$ square matrices $\{\boldsymbol{A}^{(ij)}\}_{ij}$ (where $(\boldsymbol{A}^{(ij)})^T = \boldsymbol{A}^{(ji)}$), how do we know if it exists a set of $n$ vectors $\{ \boldsymbol{v}_i \}$, that it fulfills the following set of equations $$\{\, \boldsymbol{v}_i^T\boldsymbol{A}^{(ij)}\boldsymbol{v}_j = \alpha_{ij} \,\}?$$
This corresponds to a particular instance of a non-linear system of $\frac{1}{2}n(n-1)$ equations with $n^2$ variables. The system is described by the matrices $\{\boldsymbol{A}^{(ij)} \}$ and the variables by the components of the $n$ vectors. I would like to know which conditions the matrices $\{\boldsymbol{A}^{(ij)}\}_{ij}$ must fulfill to ensure that exist at least one solution for any $\{ \alpha_{ij} \}$.