I am interested in understanding for what values of $\lambda$ is $(A - \lambda I)^{-1} b$ component-wise non-negative. Here $A$ is a $n\times n$ matrix and $b$ is a $n\times 1$ vector.
I know there is work related to this problem when $A$ and $b$ are non-negative as well as well as when $A$ is underdetermined. However, I am interested in a more general case where $A$ is an arbitrary square matrix.
I also know this is related to convex optimization and finding feasible points. However, I am not just interested in finding such a point, but characterizing all such points.