# An Introduction to Invariant Subspaces Using a Cube

An Introduction to Invariant Subspaces Using a Cube

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, . The action of the transformation can be completely understood by considering how it acts on each of three -invariant subspaces with direct sum .

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