Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 23
Xb ��I � Xk ���I � Xk � Xk 2 � Xk 3 � Xk 4 � ... � Xk n ��Xb �1 � I
Since the first factor is simply X, we finally obtain the inverse of X in terms of Xk �
X b �1 �Xs as:
X �1 � �I � Xk � Xk 2 � Xk 3 � Xk 4 � ... � Xk n ��Xb �1
Or, specifically in terms of the body and soul of X:
X �1 �
�I � �Xb �1 �Xs� � �Xb �1 �Xs� 2 � ... � �Xb �1 �Xs� n ��
Xb �1
� GrassmannMatrixInverse
13.4
13.5
This formula is straightforward to implement as a function. In GrassmannAlgebra we can
calculate the inverse of a Grassmann matrix X by using GrassmannMatrixInverse.
? GrassmannMatrixInverse
GrassmannMatrixInverse�X� computes the
inverse of the matrix X of Grassmann numbers in a
space of the currently declared number of dimensions.
A definition for GrassmannMatrixInverse may be quite straighforwardly developed
from formula 13.4.
GrassmannMatrixInverse[X_]:=
Module[{B,S,iB,iBS,K},B=Body[X];S=Soul[X];
iB=Inverse[B];iBS=�[�[iB�S]];
K=Sum[(-1)^i GrassmannMatrixPower[iBS,i],{i,0,Dimension}];
�[�[K�iB]]]
As a first simple example we take the matrix:
2001 4 26