Can the following system of equations be solved using Gaussian Elimination?
$$ \begin{bmatrix} s_{00} & s_{01} & s_{02} & s_{03}\\ s_{10} & s_{11} & s_{12} & s_{13}\\ s_{20} & s_{21} & s_{22} & s_{23}\\ s_{30} & s_{31} & s_{32} & s_{33}\\ \end{bmatrix} \begin{bmatrix} x^2_0 \\ x^2_1 \\ x^2_2 \\ x^2_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} $$
$$ x_{i} \in \mathbb{R}, s_{ij} \in \mathbb{R}, 0 \leq i \leq 3, 0 \leq j \leq 3 $$
If one were to let $$w_i = x^2_i, 0 \leq i \leq 3,$$ then the above system is (trivially) transformed to
$$ \begin{bmatrix} s_{00} & s_{01} & s_{02} & s_{03}\\ s_{10} & s_{11} & s_{12} & s_{13}\\ s_{20} & s_{21} & s_{22} & s_{23}\\ s_{30} & s_{31} & s_{32} & s_{33}\\ \end{bmatrix} \begin{bmatrix} w_0 \\ w_1 \\ w_2 \\ w_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} $$
which CAN be solved via Gaussian Elimination. Now assuming that
$$ \begin{bmatrix} w'_0 \\ w'_1 \\ w'_2 \\ w'_3 \end{bmatrix} $$
is a unique solution to the above system does that mean that
$$x_i = \pm\sqrt{w'_i}$$
is the zero-dimensional solution set for the original system? Clearly this is only the case when $$w'_i \geq 0$$.
I did try to find articles dealing with systems of quadratic forms that have a diagonal Matrix but I could not find anything relevant.
If the above approach is wrong can someone point me in the right direction?
Thank you in advance