Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 5
A � ��2 � 3e1 , e1 � 6e2 �, �5 � 9e1 � e2, e2 � e1 � e2 ��; MatrixForm�A�
2 � 3e1
�
5 � 9e1 � e2 e1 � 6e2
�
e2 � e1 � e2 � Creating matrices with variable elements
To create a matrix in which the Grassmann numbers are expressed in terms of x, y, and z,
say, you can temporarily declare the basis as {x,y,z} before generating the form with
CreateGrassmannNumber. For example:
B � Basis; DeclareBasis��x, y, z��
�x, y, z�
Z � CreateGrassmannNumber���
��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z
Here we create a vector of elements of the form Z.
CreateMatrixForm��3���Z� ��MatrixForm
� ��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z
�������
��������
��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z
� ��x ��y ��z ���x � y ��x � z ��y � z ��x � y � z �
Remember to return to your original basis.
DeclareBasis�B�
�e 1 ,e2 ,e3�
13.3 Manipulating Matrices
� Matrix simplification
Simplification of the elements of a matrix may be effected by using GrassmannSimplify.
For example, suppose we are working in a Euclidean 3-space and have a matrix B:
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