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The Eigenvalue Game

A long time ago, I spent some time working in Matlab, and I discovered a demo called the “Eigenvalue Game” hidden away in one of the toolboxes.

The game works like so: the computer picks a 2×2 matrix M (with independent, normally-distributed entries). It then draws the center of the plane ℝ². You can click at any point v, and the computer draws the vectors v and M v. The goal is to click on vectors sufficiently close to eigenvectors of M. You can play my recreation of this game below; you can refresh to play again.

An eigenvector of a matrix M is a vector v such that M v is some multiple λ v of v. So, in the game above, you want to click on the points where the output vector and input vector form a line. I found this game to give me very good intuition for eigenvectors. For example, it became very intuitive how a matrix could lack real eigenvalues: if the matrix rotates its input by a constant angle, the output and input vectors will always be offset from one another.