Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 15
Conditions on the determinant of a general matrix
Due to the non-commutativity of Grassmann numbers, the notion of determinant does not
carry over directly to matrices of Grassmann numbers. To see why, consider the diagonal
Grassmann matrix composed of 1-elements x and y:
x 0
�
0 y �
The natural definition of the determinant of this matrix would be the product of the diagonal
elements. However, which product should it be, x�y or y�x?. The possible noncommutativity
of the Grassmann numbers on the diagonal may give rise to different
products of the diagonal elements, depending on the ordering of the elements.
On the other hand, if the product of the diagonal elements of a diagonal Grassmann matrix is
unique, as would be the case for even elements, the determinant may be defined uniquely as
that product.
� Determinants of even Grassmann matrices
If a Grassmann matrix is even, its elements commute and its determinant may be defined in
the same way as for matrices of scalars.
To calculate the determinant of an even Grassmann matrix, we can use the same basic
algorithm as for scalar matrices. In GrassmannAlgebra this is implemented with the
function GrassmannDeterminant. For example:
B � EvenGrade�M�; MatrixForm�B�
� a1,1 � d1,1 e1 � e2 a1,2 � d1,2 e1 � e2
�
a2,1 � d2,1 e1 � e2 a2,2 � d2,2 e1 � e2
B1 � GrassmannDeterminant�B�
�a1,1 � d1,1 e1 � e2 ���a2,2 � d2,2 e1 � e2 � �
�a1,2 � d1,2 e1 � e2 ���a2,1 � d2,1 e1 � e2 �
Note that GrassmannDeterminant does not carry out any simplification, but instead
leaves the products in raw form to facilitate the interpretation of the process occurring. To
simplify the result one needs to use GrassmannSimplify.
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