Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 6
�������������
�3; B� ��1 � x � x, 2 � e1 ���� e2 �, �e1 � e2 ,3� x � y � z � w��;
MatrixForm�B�
� 1 � x � x 2�e1�e2 ��������������� �
e1 � e2 3 � x � y � z � w
To simplify the elements of the matrix we enter:
��B�
��1, 2 � e1 � e2 �, �e3, 3��
If we wish to simplify any orthogonal elements, we can use ToMetricForm. Since we are in
a Euclidean space the scalar product term is zero.
ToMetricForm���B��
��1, 2�, �e3 ,3��
� The grades of the elements of a matrix
The GrassmannAlgebra function Grade returns a matrix with each element replaced by a
list of the different grades of its components. Thus, if an element is composed of several
components of the same grade, only that one grade will be returned for that element. For
example, take the matrix A created in the previous section:
A �� MatrixForm
2 � 3e1 e1 � 6e2
�
�
5 � 9e1 � e2 e2 � e1 � e2
Grade�A� ��MatrixForm
�
�0, 1� 1
�0, 2� �1, 2� �
� Taking components of a matrix
We take the matrix M created in Section 13.2.
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