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 .



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+