Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 32
Hence:
� a p �L p � DiagonalMatrix��� ap �Λ1 p , � ap �Λ2 p ,…,� ap �Λm p ��
f�L� � DiagonalMatrix��f�Λ1�, f�Λ2�, …,f�Λm ���
Hence, finally we have an expression for the function f of the matrix A in terms of the
eigenvector matrix and a diagonal matrix in which each diagonal element is the function f
of the corresponding eigenvalue.
f�A� � X � DiagonalMatrix�
�f�Λ1�, f�Λ2�, …,f�Λm��� � X �1
� GrassmannMatrixFunction
13.13
We have already seen in Chapter 9 how to calculate a function of a Grassmann number by
using GrassmannFunction. We use GrassmannFunction and the formula above to
construct GrassmannMatrixFunction for calculating functions of matrices.
? GrassmannMatrixFunction
GrassmannMatrixFunction�f�A�� calculates a function f
of a Grassmann matrix A with distinct eigenvalues, where
f is a pure function. GrassmannMatrixFunction��fx,x�,A�
calculates a function fx of a Grassmann matrix A with
distinct eigenvalues, where fx is a formula with x as variable.
It should be remarked that, just as functions of a Grassmann number will have different
forms in spaces of different dimension, so too will functions of a Grassmann matrix.
� Exponentials and Logarithms
In what follows we shall explore some examples of functions of Grassmann matrices in 3space.
2001 4 26
�3; A� ��1 � x, x � z�, �2, 3 � z��; MatrixForm�A�
1 � x x�z �
2 3�z �