Grassmann Algebra

TheExteriorProduct.nb 28
Let ei
m
be basis m-elements and ei
m
be their corresponding cobasis (nÐm)-elements (i: 1,É, Ν).
�����
To fix ideas, suppose we expand a determinant about the first m rows:
�Α1 � � � Αm ���Αm�1 � � � Αn� � �a1�e1
m
Here, the ai and ai
� a2�e2
m
� � � aΝ�eΝ �
m
� �����
�
a1 ����� e1 � a2 �����
����� m
e2 � � � aΝ �����
����� m
eΝ
�����
����� m �
����� are simply the coefficients resulting from the partial expansions. As with
basis 1-elements, the exterior product of a basis m-element with the cobasis element of another
basis m-element is zero. That is:
ei
m
� ej �∆ij�e1 � e2 � � � en
����� m
Expanding the product and applying this simplification yields finally:
D � a1�a1 ����� � a2�a2 ����� � � � aΝ�aΝ �����
This is the Laplace expansion. The ai are minors and the ����� ai are their cofactors.
� Calculation of determinants using minors and cofactors
Consider the fourth order determinant:
��������
D0 � �
������� �
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
��������
�
�������
;
�
We wish to calculate the determinant using the cofactor methods above. We will use
GrassmannAlgebra to assist with the computations, so we must declare the basis to be 4dimensional
and the aij to be scalars.
�4; DeclareExtraScalars�a_ �;
Case 1: Calculation by expansion about the first row
Α1 � a11�e1 � a12�e2 � a13�e3 � a14�e4;
Α2 � Α3 � Α4 � �a21�e1 � a22�e2 � a23�e3 � a24�e4�
� �a31�e1 � a32�e2 � a33�e3 � a34�e4�
� �a41�e1 � a42�e2 � a43�e3 � a44�e4�;
Expanding this product:
2001 4 5