Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 21
If the power required is not integral and the eigenvalues are not distinct,
GrassmannMatrixPower will return a message and the unevaluated input. Let B be a
simple 2�2 matrix with equal eigenvalues 1 and 1
B � ��1, x�, �y, 1��; MatrixForm�B�
1 x
�
y 1 �
GrassmannMatrixPower�B, 1 ����
2 �
Eigenvalues ::notDistinct :
The matrix ��1, x�, �y, 1�� does not have distinct scalar eigenvalues . The
operation applies only to matrices with distinct scalar eigenvalues .
GrassmannMatrixPower���1, x�, �y, 1��, 1
����
2 �
� Integer powers of matrices with distinct eigenvalues
In some circumstances, if a matrix is known to have distinct eigenvalues, it will be more
efficient to compute powers using GrassmannMatrixFunction for the basic calculation
engine. The function of GrassmannAlgebra which enables this is
GrassmannDistinctEigenvalueMatrixPower.
? GrassmannDistinctEigenvaluesMatrixPower
GrassmannDistinctEigenvaluesMatrixPower�A,p�
calculates the power p of a Grassmann matrix A with
distinct eigenvalues. p may be either numeric or symbolic.
We can check whether a matrix has distinct eigenvalues with DistinctEigenvaluesQ.
For example, suppose we wish to calculate the 100th power of the matrix A below.
2001 4 26
A �� MatrixForm
1 x
�
y 2 �
DistinctEigenvaluesQ�A�
True