Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 29
where As
1
Ab and As
1
, Xs , and Ls
1
1
Ab � Xs
1
� As
1
� Xb � Xb � Ls
1
� Xs
1
� Lb
13.10
are the components of As , Xs , and Ls of grade 1. Here we know
and have already solved for Xb and Lb , leaving just Xs
1
and Ls to be determined.
1
However, although this process would be helpful if trying to solve by hand, it is not
necessary when using the Solve function in Mathematica.
� GrassmannMatrixEigensystem
The function implemented in GrassmannAlgebra for calculating the eigensystem of a matrix
of Grassmann numbers is GrassmannMatrixEigensystem. It is capable of calculating the
eigensystem only of matrices whose body has distinct eigenvalues.
? GrassmannMatrixEigensystem
GrassmannMatrixEigensystem�A� calculates a list comprising the
matrix of eigenvectors and the diagonal matrix of eigenvalues
for a Grassmann matrix A whose body has distinct eigenvalues.
If the matrix does not have distinct eigenvalues, GrassmannMatrixEigensystem will
return a message telling us. For example:
A � ��2, x�, �y, 2��; MatrixForm�A�
2 x
�
y 2 �
GrassmannMatrixEigensystem�A�
Eigenvalues ::notDistinct :
The matrix ��2, x�, �y, 2�� does not have distinct scalar eigenvalues . The
operation applies only to matrices with distinct scalar eigenvalues .
GrassmannMatrixEigensystem���2, x�, �y, 2���
If the matrix has distinct eigenvalues, a list of two matrices is returned. The first is the
matrix of eigenvectors (whose columns have been normalized with an algorithm which tries
to produce as simple a form as possible), and the second is the (diagonal) matrix of
eigenvalues.
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