Let us assume that a vector $x \in \mathbb{R}^n$ is given and we are looking for a matrix $A \in \mathbb{R}^{n\times n}$ which yields $A x = x$. That is, we perform a sort of reverse questioning: Instead of having a given $A$ and an unknown $x$, the vector $x$ is known but $A$ not.

Is there any theory which discusses such sort of equations? Pardon me if I am missing something completely trivial, but it would be great if someone could give me a hint.

Best wishes,
Andreas