Suppose we have unknown scalars $x_1, x_2, ...,x_m \in \mathbb{R}$, known matrices $A_1, A_2, ...,A_m \in \mathbb{R}^{n\times n}$, and two known vectors $s_0, s_1\in\mathbb{R}^n$.
I want to find $x_1, x_2, ...,x_m$ that satisfy the following equation:
$$s_1 = \exp\!\left(\sum_{i=1}^{m}x_iA_i\right)\!s_0$$
I know that there can be no solutions, a finite number of solutions, or an infinite number of solutions. I would like to find all of the solutions that exist. The matrices $A_i$ don't have a special form or property. If you have recommended books on the topic that would be extremely helpful. I am also interested in ways to numerically find all of the solutions.