The reason we are stressing eigenvectors and their importance is because the action of a matrix $A$ on one of its eigenvectors $v$ is
Of course, not every vector can be ... BUT ... For certain matrices (including ones with distinct eigenvalues, $\lambda $'s), their eigenvectors span ${\mathbb{C}}^{n}$, meaning that for any $x\in {\mathbb{C}}^{n}$, we can find $\left\{{\alpha}_{1},{\alpha}_{2},{\alpha}_{n}\right\}\in \mathbb{C}$ such that:
$$x=\sum _{i}^{}{\alpha}_{i}{v}_{i}$$ $$b=\sum _{i}^{}{\alpha}_{i}{\lambda}_{i}{v}_{i}$$ The LTI system above represents our Equation 1. Below is an illustration of the steps taken to go from $x$ to $b$. $$x\to (\alpha ={V}^{-1}x)\to (\Lambda {V}^{-1}x)\to (V\Lambda {V}^{-1}x=b)$$ where the three steps (arrows) in the above illustration represent the following three operations: