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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.

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