eigenshow()

Demonstrator of geometric intuition behind eigenvectors and singular vectors.

A figure opens showing a vector 𝑥 on the unit circle and its image 𝐴𝑥 via a given 2x2 matrix 𝐴. As you move the mouse around the circle, the image vectors trace out an ellipse. Click the mouse to leave a marker for the current source and image vectors.

An eigenvector occurs when 𝐴𝑥 and 𝑥 are parallel, and the associated eigenvalue is the multiplier. When the toggle is moved to select "svd", then the images of two vectors 𝑥 and 𝑦 are shown while 𝑥 and 𝑦 are kept perpendicular. When the image vectors are also perpendicular, then you are seeing all of the left and right singular vectors.

The left panel includes a selector of different matrices. Some things to observe: Is the number of eigenvectors (not counting trivial sign flips) the same in all cases? What about the singular vectors? Does either set of vectors have any correspondence to the image ellipse?