Home  18.013A  Chapter 32 


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 $\stackrel{\u27f6}{v}$ and use the mmult command (or do it out yourself) to compute $M\stackrel{\u27f6}{v}$ and for each component of $\stackrel{\u27f6}{v}$ compute the ratio of $\frac{{(Mv)}_{i}}{{v}_{i}}$ .
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 $\stackrel{\u27f6}{v}$ ; then they take turns modifying $\stackrel{\u27f6}{v}$ 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.
