Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 16
��B1�
�a 1,2 a 2,1 � a 1,1 a 2,2 � �a 2,2 d 1,1 � a 2,1 d 1,2 � a 1,2 d 2,1 � a 1,1 d 2,2 � e 1 � e 2
13.6 Matrix Powers
� Positive integer powers
Powers of matrices with no body
Positive integer powers of a matrix of Grassmann numbers are simply calculated by taking
the exterior product of the matrix with itself the requisite number of times - and then using
GrassmannSimplify to compute and simplify the result.
We begin in a 4-space. For example, suppose we let
�4; A� ��x, y�, �u, v��; MatrixForm�A�
x y
�
u v �
The second power is:
��A � A�
��x � x � y � u, x � y � y � v�, �u � x � v � u, u � y � v � v��
When simplified this becomes:
A 2 � ����A � A��; MatrixForm�A 2 �
�
��u � y� ��v � y� � x � y
�
��u � v� � u � x u�y We can extend this process to form higher powers:
2001 4 26
A3 � ����A � A2 ��; MatrixForm�A3 �
��u � v � y� � 2u� x � y v�x�y �
��u � v � x� �2u� v � y � u � x � y �