I am faced with the following problem and unsure how to proceed.
Given functions ${\bf M}_1, {\bf M}_2, \dots, {\bf M}_n : {\mathbb R}^n \to {\mathbb R}^{n \times n}$ and a vector ${\bf y} \in {\mathbb R}^n$, define $$ f_i ({\bf x}) := {\bf y}^\top {\bf M}_i({\bf x}) \, {\bf y} $$
and ${\bf f} := \begin{bmatrix} f_1 & f_2 & \dots & f_n \end{bmatrix}^\top$. I would like find the vector field fixed point of $\bf f$ via $ {\bf x} = {\bf f} ({\bf x}) $, or, alternatively, via the seemingly weaker condition
$${\bf x} \propto \sum_{i=1}^n f_i({\bf x}) {\bf e}_i$$
where ${\bf e}_i$ is a vector of zeros with a $1$ in the $i$-th entry.
The function ${\bf M}$ can be defined so that its output is a symmetric matrix, if that helps.
This has the flavor of a system of equations or an eigenvalue/eigenvector equation, but with some obvious differences.
Is this problem well-posed? Has anyone seen a problem like this and might know how to go about characterizing the solution? Or otherwise, is there another way to write the problem that might help me see where to start?