An Introduction to Invariant Subspaces Using a Cube

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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 .

Contributed by: Keith Schneider (September 2007)
Open content licensed under CC BY-NC-SA



Let be a vector in , where each component is the value on a face of the cube. It is easy to define the linear transformation that represents a turn. The vector space can be decomposed into three -invariant subspaces: = {| all faces have the same value}, ={| the sum of all faces is zero and opposite faces have the same value}, and ={| the sum of all faces is zero and opposite faces also sum to zero}. Examining the action of on each subspace makes the reasons for the convergence to the average value clear.

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