2010-05-28

Skewed Commutative (?) Matrix (4)

Marco's Question
In the conference, I met my old friend, Marco. This conference is the state of the art computer graphics conference, so, the story of 2 by 2 matrix is totally not be interedted in. However, he asked me what I am on, then I answer the matrix story. Then he asked me a question, ``Can we visualize such matrices?'' It is a brilliant question. What is the common property of these matrix? Is that any common geometrical property, for instance?

The geometric property of my first answer, Equation(1) of Skewed Commutative (?) Matrix (1), is 90 degree rotation (A) and flip y direction (B). In Figure AB_BA, these matrices applied to the vector (1,0) which is shown as the red arrow.

Figure AB_BA

As shown in Figure AB_BA, when AB is applied to the vector (1 0)^T, first B reflects Y axis direction, but the vector (1 0)^T has no change since Y component is 0. Then A rotates the vector by 90 degrees. The result is a Y axis + direction vector, (0 1)^T. Instead, BA first rotate the vector with 90 degrees by A, (1 0)^T becomes (0 1)^T, then Y axis reflection makes (0 -1)^T. In the end, these two operation, AB and BA are negation relationship, AB = -BA. This is also one of the good intuitive example of AB != BA (!= means not equal here).  This property of matrix operation is non-commutative. The multiplication of AB is

Result AB and BA

Therefore, AB = -BA. My question is what the other pairs of matrix means, and, Marco's question is can we visualize them? There are 56 pairs of matrix that satisfy the condition. You can find all the matrix pairs in the Appendix.