Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 17
A3 � ��A � A2 �; MatrixForm�A3�
��u � v � y� � 2u� x � y v�x�y �
��u � v � x� �2u� v � y � u � x � y �
Because this matrix has no body, there will eventually be a power which is zero, dependent
on the dimension of the space, which in this case is 4. Here we see that it is the fourth power.
A4 � ����A � A3 ��; MatrixForm�A4 �
0 0
�
0 0 �
Powers of matrices with a body
By giving the matrix a body, we can see that higher powers may no longer be zero.
B � A � IdentityMatrix�2�; MatrixForm�B�
1 � x y
�
u 1�v �
B 4 � ����B � ��B � ��B � B����; MatrixForm�B 4 �
�
1 � 4x� 6u� y � 4u� v � y � 8u� x � y 4 y� 6v� y � 6x� y � 4v� x � y
4u� 6u� v � 6u� x � 4u� v � x 1� 4v� 6u� y � 8u� v � y � 4u� x
GrassmannIntegerPower
Although we will see later that the GrassmannAlgebra function GrassmannMatrixPower
will actually deal with more general cases than the positive integer powers discussed here,
we can write a simple function GrassmannIntegerPower to calculate any integer power.
Like GrassmannMatrixPower we can also easily store the result for further use in the
same Mathematica session. This makes subsequent calculations involving the powers
already calculated much faster.
GrassmannIntegerPower[M_,n_?PositiveIntegerQ]:=
Module[{P},P=NestList[�[�[M�#]]&,M,n-1];
GrassmannIntegerPower[M,m_,Dimension]:=First[Take[P,{m}]];
Last[P]]
As an example we take the matrix B created above.
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