The way I understand the concept of the eigenvectors of a $n$ x $n$ matrix $A$ is from the effect of $A$ on the vectors: If $\mathbf{b}$ is an eigenvector of $A$, then the only effect of $A$ on $\mathbf{b}$ is stretching.
I am looking for another viewpoint, if exists, from the linear combination definition of matrix-vector multiplication: the entries of $\mathbf{b}$ are the weights of the column vectors of $A$ in their combination. Thus, we see that using the entries of an eigenvector of a matrix as weights in a linear combination of the column vectors of that matrix results in a vector that is a scaling version of the vector consisting of the weights. What is the significance of this observation, if any?