Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 13
13.5 The Transpose
Conditions on the transpose of an exterior product
If we adopt the usual definition of the transpose of a matrix as being the matrix where rows
become columns and columns become rows, we find that, in general, we lose the relation for
the transpose of a product being the product of the transposes in reverse order. To see under
what conditions this does indeed remain true we break the matrices up into their even and
odd components. Let A � Ae � Ao and B � Be � Bo , where the subscript e refers to the even
component and o refers to the odd component. Then
�A � B� T � ��Ae � Ao ���Be � Bo �� T � �Ae � Be � Ae � Bo � Ao � Be � Ao � Bo � T �
�Ae � Be� T � �Ae � Bo � T � �Ao � Be� T � �Ao � Bo � T
B T � A T � �Be � Bo � T ��Ae � Ao � T �
�B e T � Bo T ���Ae T � Ao T � � Be T � Ae T � Be T � Ao T � Bo T � Ae T � Bo T � Ao T
If the elements of two Grassmann matrices commute, then just as for the usual case, the
transpose of a product is the product of the transposes in reverse order. If they anticommute,
then the transpose of a product is the negative of the product of the transposes in
reverse order. Thus, the corresponding terms in the two expansions above which involve an
even matrix will be equal, and the last terms involving two odd matrices will differ by a sign:
Thus
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�Ae � Be � T � B e T � Ae T
�Ae � Bo � T � B o T � Ae T
�Ao � Be � T � B e T � Ao T
�Ao � Bo � T ���B o T � Ao T �
�A � B� T � B T � A T � 2��B o T � Ao T � �
B T � A T � 2��Ao � Bo� T
13.1