Eigenvectors and Eigenvalues explained visually

By Victor Powell and Lewis Lehe Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive. To begin, let $v$ be a vector (shown as a point) and $A$ be a matrix with columns $a_1$ and $a_2$ (shown as arrows).