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An Introduction to Invariant Subspaces Using a Cube

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turns
1
back = 4.5000
front = 4.5000
left = 3.2500
right = 3.2500
bottom = 2.7500
top = 2.7500
The numbers 1 through 6 are placed on the faces of a cube. At every turn the number on each face is replaced by the average of its four adjacent faces. The value of each face is displayed as a color. The values converge quickly to the average of the initial values of all faces. Think of the collection of values as a six-dimensional vector being acted on at each turn by a linear transformation,
T
. The action of the transformation can be completely understood by considering how it acts on each of three
T
-invariant subspaces with direct sum
6
.
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