2x2 Matrix Visualization

A =

λ1 =
λ2 =
x1 =

x2 =



I've commonly seen matrices in textbooks represented in two dimensions as ellipses. This page is my attempt at making that representation interactive. Although this is limited to 2x2 matrices, the same concepts can be extrapolated to higher dimensions.

Here, the 2x2 identity matrix is represented by a two-dimensional unit ball centered at the origin, as shown by the yellow circle in the above plot, which is a representation of \lVert \mathbf{A} \rVert_2 . This unit disk is mapped to an ellipse by the transformation that matrix A represents.

λ1 and λ2 are the eigenvalues of matrix A, and x1 and x2 are the corresponding eigenvectors of matrix A. These are represented visually by the green and blue vectors on the above plot.



There are quite a few cool observations about this visualization. Some day, when I have more time, I will write more about them. But for now, here is a short list of observations. ... and there are probably a lot more things that I've overlooked, but that's all I've come up with for now. Please let me know if you've come up with anything else, or if you have any comments, questions, or corrections!

Source Code