|
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 Mv and for each component of v compute the
ratio of  .
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.
|
