Grassmann Algebra

ExplorGrassmannMatrixAlgebra.nb 20
Let A be the 2�2 matrix which differs from the matrix discussed above by having distinct
eigenvalues 1 and 2. We will discuss eigenvalues in Section 13.9 below.
A � ��1, x�, �y, 2��; MatrixForm�A�
1 x
�
y 2 �
We compute the square root of A.
sqrtA � GrassmannMatrixPower�A, 1
����
2 �
��1 � �� 3
���� �
2 �����
2 � x � y, ��1 � �����
2 � x�,
���1 � �����
2 � y, �����
2 � 1 ���� ��4 � 3
4 �����
2 � x � y��
To verify that this is indeed the square root of A we 'square' it.
����sqrtA � sqrtA��
More generally, we can extract a symbolic pth root. But to have the simplest result we need
to ensure that p has been declared a scalar first,
DeclareExtraScalars��p��
�a, b, c, d, e, f, g, h, p, �, �_ � _�? InnerProductQ, _� 0
pthRootA � GrassmannMatrixPower�A, 1 ����
p �
��1 � ��1 � 2 1
����
p � 1
����
���1 � 2 1
����
p � x � y, ��1 � 2 1
p � y, 2 1
���� p � �
1 �����
����
�1 � 2
�
p �
����
p � x�,
�
��������������
����� x � y��
p
�
1
����
2�1� p
We can verify that this gives us the previous result when p = 2.
2001 4 26
sqrtA � pthRootA �. p� 2 �� Simplify
True