Grassmann Algebra

TheInteriorProduct.nb 16
Thus Α 3 ���� Β 2
can be computed by entering:
MatrixProduct�Β ���� Α�. Α
2 2 1
�Β 2
� Α2 � Α3� Α1 � �Β � Α1 � Α3� Α2 � �Β
2
2
6.5 The Inner Product
Inner products of general elements
� Α1 � Α2� Α3
As we have already discussed in Section 6.3 the interior product of two simple elements
Α �Α1� Α2 � � � Αm and Β �Β1� Β2 � � � Βm of the same grade is called more
m m
specifically the inner product of Α and Β. In what follows we derive a simple expression for this
m m
often-encountered quantity. We begin with the form we wish to re-express.
Α m ���� Β m
� �Α1 � Α2 � � � Αm � ���� �Β1 � Β2 � � � Βm �
From formula 6.21 for extended interior products:
Α ����
m ����
Β � Β
�k1
k2
�
� � � Β
kp�
��� � ���Α m ���� Β k1
� ���� Β � ���� �� ���� Β
k2
kp
we can write �Α1 � Α2 � � � Αm� ���� �Β1 � Β2 � � � Βm� as:
��Α1 � Α2 � � � Αm� ���� Β1� ���� �Β2 � � � Βm �
or, by rearranging the Β factors to bring Βj to the beginning of the product as:
��Α1 � Α2 � � � Αm� ���� Βj� ���� ���1� j�1 ��Β1 � Β2 � � � ���� j � � � Βm ��
The Interior Common Factor Theorem gives the first factor of this expression as:
�Α1 � Α2 � � � Αm� ���� Βj � �
m
i�1
Thus, �Α1 � Α2 � � � Αm � ���� �Β1 � Β2 � � � Βm � becomes:
�Αi ���� Βj ����1� i�1 �Α1 � Α2 � � � ���� i � � � Αm
�Α1 � Α2 � � � Αm � ���� �Β1 � Β2 � � � Βm � �
m
� �Αi ���� Βj ����1�
i�1
i�j ��Α1 � � � ���� i � � � Αm � ����
�Β1 � � � ���� j � � � Βm �
If this process is repeated, we find the strikingly simple formula:
2001 4 5
6.35