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Home | 18.013A | Chapter 32 |
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Here is a game you can set up on a spreadsheet. Enter an arbitrary matrix somewhere.
Three by three is a good way to start.
Enter a 3 component column vector and use the mmult command (or do it out yourself) to compute and for each component of 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 and ; then they take turns modifying 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 .
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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