Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 19
GrassmannMatrixPower�B, 4�
��1 � 4x� 6u� y � 4u� v � y � 8u� x � y, 4 y � 6v� y � 6x� y � 4v� x � y�,
�4 u� 6u� v � 6u� x � 4u� v � x, 1 � 4v� 6u� y � 8u� v � y � 4u� x � y��
� Negative integer powers
From this point onwards we will be discussing the application of the general
GrassmannAlgebra function GrassmannMatrixPower.
We take as an example the following simple 2�2 matrix.
A � ��1, x�, �y, 1��; MatrixForm�A�
1 x
�
y 1 �
The matrix A �4 is calculated by GrassmannMatrixPower by taking the fourth power of the
inverse.
iA4 � GrassmannMatrixPower�A, �4�
��1 � 10 x � y, �4 x�, ��4 y,1� 10 x � y��
We could of course get the same result by taking the inverse of the fourth power.
GrassmannMatrixInverse�GrassmannMatrixPower�A, 4��
��1 � 10 x � y, �4 x�, ��4 y,1� 10 x � y��
We check that this is indeed the inverse.
����iA4 � GrassmannMatrixPower�A, 4���
��1, 0�, �0, 1��
� Non-integer powers of matrices with distinct eigenvalues
If the matrix has distinct eigenvalues, GrassmannMatrixPower will use
GrassmannMatrixFunction to compute any non-integer power of a matrix of Grassmann
numbers.
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