Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 14
�A � B� T � B T � A T � Ao � Bo � 0
� Checking the transpose of a product
13.2
It is useful to be able to run a check on a theoretically derived relation such as this. To
reduce duplication effort in the checking of several cases we devise a test function called
TestTransposeRelation.
TestTransposeRelation[A_,B_]:=
Module[{T},T=Transpose;
�[T[�[A�B]]]� �[�[T[B]�T[A]]-2�[T[OddGrade[B]]�T[OddGrade[A]]]]]
As an example, we test it for two 2�2 matrices in 2-space:
A �� MatrixForm
2 � 3e1 e1 � 6e2
�
�
5 � 9e1 � e2 e2 � e1 � e2
M �� MatrixForm
� a1,1 � e1 b1,1 � e2 c1,1 � d1,1 e1 � e2 a1,2 � e1 b1,2 � e2 c1,2 � d1,2 e1 � e2
�
a2,1 � e1 b2,1 � e2 c2,1 � d2,1 e1 � e2 a2,2 � e1 b2,2 � e2 c2,2 � d2,2 e1 � e2
TestTransposeRelation�M, A�
True
It should be remarked that for the transpose of a product to be equal to the product of the
transposes in reverse order, the evenness of the matrices is a sufficient condition, but not a
necessary one. All that is required is that the exterior product of the odd components of the
matrices be zero. This may be achieved without the odd components themselves necessarily
being zero.
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