I'm looking for a direction to solve the following non-linear equation system (solving for $X$):
We have vector $\textbf{X}=\{X_1,\dots, X_t\}$ and the following system: $$X_i = \frac{1}{\sum_{j}^{T}X_j\beta_{ij}}$$ where $\beta_{ij}=\beta_{ji}$ and $\beta$ is a known matrix.
I start to look on specific case where $\textbf{X}\in\mathbb{R}^2$. The solution for this case is: $$\frac{X_1^2}{X_2^2}=\frac{\beta_{22}}{\beta_{11}}$$
I wasn't able to generalize this result to the general case of $\space\textbf{X}\in\mathbb{R}^n$. If it helps we can assume $\sum{X_i}=1$ and $X_i \geq 0 \space$.
Thanks in advance.