# A real system of bilinear equations with $2n$ unknown and equations

I have the following system of $$2n$$ bilinear equations, for a square invertible matrix $$A \in \mathbb{R}_{n \times n}$$, and $$2n$$ unknowns organized in vectors $$x,y \in \mathbb{R}^n$$: $$diag(y) A x = {\mathbf 1}_n$$ $$diag(x) A^T y = {\mathbf 1}_n$$ where $$diag(x)$$ denotes a diagonal matrix with diagonal elements $$x$$, and $${\mathbf 1}_n$$ is a vector of ones of length $$n$$.

1. What can be said about the solution of this system? for example, when is there a (unique?) solution? While in general solving bilinear equations is hard, this system has a specialized simple symmetric structure that can be used.

2. We can easily implement an iterative algorithm with iterations: $$x^{(t+1)} = (diag(y^{(t)})A)^{-1} {\mathbf 1}_n$$ $$y^{(t+1)} = (diag(x^{(t)})A^T)^{-1} {\mathbf 1}_n.$$

What can be said about the convergence of this algorithm? is there a better approach? any hope for a closed-form solution?