EigenShow.jl

Interactive demonstrator of eigenvectors and singular vectors for Julia

Usage

1. Install Julia. Version 1.7 or greater is recommended.

2. Start Julia and enter the command

   ]add EigenShow

   It may take a few minutes to download and install the necessary packages.

3. Enter the commands

   using EigenShow
   eigenshow()

   (The using command is needed only once per Julia session.) After 20 seconds or so, a new window should open with the demonstrator. It runs independently and can simply be closed when you are done with it.

In EVD mode. Initially you see a vector $x$ on the unit circle and, in a different color, the vector $Ax$ resulting from matrix-vector multiplication using the current matrix $A$ chosen in the selection box. As you move the mouse, $x$ moves around the unit circle and $Ax$ traces out an ellipse (or, in a degenerate case, a line segment). When $Ax$ is parallel to $x$, then $x$ is an eigenvector of $A$, and the (signed) ratio of the vector lengths is the associated eigenvalue. When $x$ is an eigenvector, so is $-x$. The matrix may have zero, one, or two distinct real eigenvectors.

In SVD mode. As you move the mouse, a perpendicular pair of vectors $x,y$ move around the unit circle. In a different color, you also see $Ax$ and $Ay$ for the current matrix $A$ chosen in the selection box. When $Ax$ and $Ay$ are perpendicular to each other, then $x$ and $y$ are right singular vectors of $A$, $Ax$ and $Ay$ are left singular vectors of $A$, and the (unsigned) ratios of lengths of $Ax$ to $x$ and $Ay$ to $y$ are associated singular values. Aside from the trivial duplications $x ↦ -x$ and $y ↦ -y$, every real $2\times 2$ matrix has a unique pair of real right singular vectors.

In either mode, you can click the mouse button to mark a point for future reference.

Acknowledgement

This function is inspired by EIGSHOW.M, which is held in copyright by The MathWorks, Inc and found at: Cleve Moler (2021), Cleve_Lab, MATLAB Central File Exchange. Retrieved October 25, 2021.