I've been playing around--for longer than I care to admit--on this amazing site that allows you to visualize matrix transformations, determinants, and eigenvectors in order to develop an intuition of how they work. I've gotten a great intuition of matrix transformations and determinants, but I'm having a really hard time with eigenvectors. I understand the computation just fine. I also understand the "words" explaining the direction concept just fine. I even see that when I draw a vector that aligns with an eigenvector, that vector's direction doesn't change after the transformation. However, I'm just not getting the visceral understanding of the positioning of the eigenvectors and how they relate to the transformed basis. My gut is telling me that the transformed basis vectors define some "maximal direction of shear" and this can be used to find the direction of the eigenvectors.
QUESTION: Is there a geometrical construction or (even better) a way to "eyeball" the eigenvectors when you know the transformed basis vectors? An intuitive explanation (along with a drawing) for the non-mathematician is appreciated.
ASIDE: One thing that I have noticed is that when I keep $j$ fixed and increase the magnitude of $i$, one of the eigenvectors seems to be "pulled" towards the $i$. The other eigenvector always points in the direction of $j$. Thus, there seems to be some interplay between the relative magnitudes of the transformed basis vectors and the angle between the original basis and the transformed basis.
I left $j$ unchanged. I systematically altered $i$ to look for patterns in how eigenvectors change w.r.t changes in $i$. One eigenvector always remained parallel to $j$, so I only examined the effect on the other.
I first rotated $i$ from $0$ to $90$ degrees and found that the eigenvector sweeps an arc of a circle from $-90$ to $-45$ degrees.
Next, I scaled $i$ by varying factors from $0.5$ to $1000$ and rotated it as before after each scaling. I found some interesting patterns that I'm still pondering on.
Perhaps this will stimulate some thoughts.