]> 32.11 Guessing Eigenvectors

32.11 Guessing Eigenvectors

Here is a game you can set up on a spreadsheet. Enter an arbitrary matrix $M$ somewhere.

Three by three is a good way to start.

Enter a 3 component column vector $v ⟶$ and use the mmult command (or do it out yourself) to compute $M v ⟶$ and for each component of $v ⟶$ compute the ratio of $( M v ) i v i$ .

Calculate the variance of these ratios (that is, the sum of their squares minus the square of their sum).

The players can take turns generating the original $M$ and $v ⟶$ ; then they take turns modifying $v ⟶$ by changing one of its components.

If the variance of the ratios decreases the player scores a point, otherwise loses one. The game ends when the variance becomes negligible, say less than $10 − 10$ .

The ratios then will be more or less the same and hence the eigenvalue associated with the eigenvector produced.

If you get too good at this, you can try with a 5 by 5 matrix, though it is boring to enter one at the start.