I was looking at an interesting visualization of eigenvalues and eigenvectors. The picture is below. I understand the meaning and computation of eigenvalues and eigenvectors, but I think I learned about them in a much more algebra forward way instead of a more graphical way. Hence my question.
Say I have a square 3x3 matrix $A$ that is made up of component vectors $a_1, a_2, a_3$. And this matrix has eigenvectors $v_1, v_2, v_3$ that correspond to 3 eigenvalues. I was wondering what the mathematical relationship was between the directions of $a_i$ and the corresponding eigenvectors of the matrix $A$. From the image it seems like the eigenvector--at least in this case, is a bisector between say $a_1, a_2$.
I am not sure if that holds in all or most cases. But seems like there should be something here related to the dot product between the $a_i$ vectors--which gives the angle between the vectors, and then the corresponding angle between each $a_i$ and the eigenvectors.