Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 33
We first calculate the exponential of A.
expA � GrassmannMatrixFunction�Exp�A��
�����x � 1
���� � ��3 ��
2 2 � x � z, 1
���� � ��1 ��
2 2 � x � z�,
����� 3 � 1
���� � ��3 ��
2 2 � z
� x � �������� �
2 �3 z
����������� � 3 � x � z,
2
� 3 �� 3 z � 1
���� ����
2 3 � x � z��
We then calculate the logarithm of this exponential and see that it is A as we expect.
GrassmannMatrixFunction�Log�expA��
��1 � x, x � z�, �2, 3 � z��
� Trigonometric functions
We compute the square of the sine and the square of the cosine of A:
s2A � GrassmannMatrixFunction��Sin�x�^2, x�, A�
��2 xCos�1� Sin�1� � Sin�1� 2 � 1
���� Sin�1� ��4 Cos�1� � Sin�3� � Sin�5�� x � z,
2
Cos�2� Sin�2� 2 x � z�,
��z Sin�2� � 2zCos�4� Sin�2� � 1
���� x Sin�2���2 � Sin�4�� � Sin�2� Sin�4� �
2
1
���� z Sin�2� Sin�4� �
2 1
���� Sin�2� �6� 6 Cos�4� � 3 Sin�4�� x � z,
2
Sin�3� 2 � z Sin�6� � 1 ���� ��Cos�2� � Cos�6� � 4 Sin�6�� x � z��
4
c2A � GrassmannMatrixFunction��Cos�x�^2, x�, A�
��Cos�1� 2 � 2xCos�1� Sin�1� � 1 ���� Cos�1���Cos�3� � Cos�5� � 4 Sin�1�� x � z,
2
�Cos�2� Sin�2� 2 x � z�,
�z Sin�2� � 2zCos�4� Sin�2� � 1
���� x Sin�2� ��2 � Sin�4�� � Sin�2� Sin�4� �
2
1
���� z Sin�2� Sin�4� �
2 1
���� Sin�2� �6� 6 Cos�4� � 3 Sin�4�� x � z,
2
Cos�3� 2 � z Sin�6� � 1
���� ��Cos�2� � Cos�6� � 4 Sin�6�� x � z��
4
and show that sin 2 x + cos 2 x = 1, even for Grassmann matrices:
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