Grassmann Algebra

TheRegressiveProduct.nb 27
Let Α m be the simple element to be factorized. Choose first an (nÐm)-element Β
whose product
n�m
with Α is non-zero. Next, choose a set of m independent 1-elements Βj whose products with
m
Β are also non-zero.
n�m
A factorization Αj of Α m is then obtained from:
Α m � a Α1 � Α2 � � � Αm
Αj �Α��Βj � Β �
m n�m
The scalar factor a may be determined simply by equating any two corresponding terms of the
original element and the factorized version.
Note that no factorization is unique. Had different Βj been chosen, a different factorization
would have been obtained. Nevertheless, any one factorization may be obtained from any other
by adding multiples of the factors to each factor.
3.37
If an element is simple, then the exterior product of the element with itself is zero. The
converse, however, is not true in general for elements of grade higher than 2, for it only requires
the element to have just one simple factor to make the product with itself zero.
If the method is applied to the factorization of a non-simple element, the result will still be a
simple element. Thus an element may be tested to see if it is simple by applying the method of
this section: if the factorization is not equivalent to the original element, the hypothesis of
simplicity has been violated.
Example: Factorization of a simple 2-element
Suppose we have a 2-element Α which we wish to show is simple and which we wish to
2
factorize.
Α 2 � v � w � v � x � v � y � v � z � w � z � x � z � y � z
There are 5 independent 1-elements in the expression for Α: v, w, x, y, z, hence we can choose
2
n to be 5. Next, we choose Β (= Β) arbitrarily as x�y�z, Β1 as v, and Β2 as w. Our two
n�m 3
factors then become:
Α1 �Α��Β1 � Β� �Α��v� x � y � z�
2 3 2
Α2 �Α��Β2 � Β� �Α��w � x � y � z�
2 3 2
In determining Α1 the Common Factor Theorem permits us to write for arbitrary 1-elements Ξ
and Ψ:
�Ξ � Ψ���v � x � y � z� � �Ξ � v � x � y � z��Ψ��Ψ � v � x � y � z��Ξ
Applying this to each of the terms of Α 2 gives:
2001 4 5
Α1 ���w � v � x � y � z��v � �w � v � x � y � z��z � v � z