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?


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