Grassmann Algebra

TheInteriorProduct.nb 38
Or, more succinctly:
�
�Α � x
� 2 �
� � �Α ���� x
� 2
�
m
m
Similarly, formula 6.98 reduces to a more obvious decomposition for x in terms of a unit melement.
� 1
� � m�1 � �
x � �Α � x� ���� Αm � ��1� �Αm ���� �Αm ���� x�
m
6.102
6.103
Because of its geometric significance when Α is simple, the properties of this equation are worth
m
investigating further. It will be shown that the square (scalar product with itself) of each of its
terms is the corresponding term of formula 6.100. That is:
� � � � � �
��Α � x� ���� Αm � ���� ��Αm � x� ���� Αm � � �Αm � x� ���� �Αm � x�
m
� � � � � �
�Α ���� �Αm ���� x�� ���� �Αm ���� �Αm ���� x�� � �Αm ���� x� ���� �Αm ���� x�
m
Further, by taking the scalar product of formula 6.102 with itself, and using formulae 6.100,
6.103, and 6.104, yields the result that the terms on the right-hand side of formula 6.102 are
orthogonal.
� � � �
��Α � x� ���� Αm � ���� �Αm ���� �Αm ���� x�� � 0
m
It is these facts that suggest the name triangle formulae for formulae 6.101 and 6.102.
Diagram of a vector x decomposed into components in and orthogonal to Α m .
The measure of the triangle components
�
Let Α �Α1� � � Αm , then:
m
� � � � m � � � �
��Α � x� ���� Αm � ���� ��Αm � x� ���� Αm � � ��1� ���Αm � x� ���� ��Αm � x� ���� Αm �� ���� Αm
m
Focussing now on the first factor of the inner product on the right-hand side we get:
2001 4 5
� � � � �
�Α � x� ���� ��Αm � x� ���� Αm � � �Α1 � � � Αm � x� ���� ��Α � x� ���� Αm �
m
m
6.104
6.105
6.106