Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 22
A
��1, x�, �y, 2��
GrassmannDistinctEigenvaluesMatrixPower�A, 100�
��1 � 1267650600228229401496703205275 x � y,
1267650600228229401496703205375 x�,
�1267650600228229401496703205375 y,
1267650600228229401496703205376 �
62114879411183240673338457063425 x � y��
13.7 Matrix Inverses
A formula for the matrix inverse
We can develop a formula for the inverse of a matrix of Grassmann numbers following the
same approach that we used for calculating the inverse of a Grassmann number. Suppose I
is the identity matrix, Xk is a bodiless matrix of Grassmann numbers and Xk i is its ith
exterior power. Then we can write:
�I � Xk ���I � Xk � Xk 2 � Xk 3 � Xk 4 � ... � Xk q � � I � Xk q�1
Now, since X k has no body, its highest non-zero power will be equal to the dimension n of
the space. That is Xk n�1 � 0. Thus
�I � Xk���I � Xk � Xk 2 � Xk 3 � Xk 4 � ... � Xk n � � I
13.3
We have thus shown that for a bodiless Xk , �I � Xk � Xk 2 � Xk 3 � Xk 4 � ... � Xk n � is the
inverse of �I � Xk�.
If now we have a general Grassmann matrix X, say, we can write X as X � Xb � Xs , where
Xb is the body of X and Xs is the soul of X, and then take Xb out as a factor to get:
X � Xb � Xs � Xb ��I � Xb �1 �Xs � � Xb ��I � Xk �
Pre- and post-multiplying the equation above for the inverse of �I � Xk�by Xb and
Xb �1 respectively gives:
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