Grassmann Algebra

TheInteriorProduct.nb 4
An alternative definition
The definition 6.1 above is chosen rather than
�����
Α � Βk (k ³ m) as a possible alternative since,
m
given the way in which the complement has been defined earlier, only the proposed definition
leads to the interior product of an m-element with itself in Euclidean space (gij �∆ij ) being
invariably positive, independent of the grade of the element or the dimension of the space. We
can see this from equation 5.69 in the last chapter. Instead of ei � 1, we would have
rather ei
m
���� ei
m
Historical Note
� ��1� m��n�m� .
Grassmann and workers in the Grassmannian tradition define the interior product of two
elements as the product of one with the complement of the other, the product being either
exterior or regressive depending on which interpretation produces a non-zero result.
Furthermore, when the grades of the elements are equal, it is defined either way. This definition
involves the confusion between scalars and n-elements discussed in Chapter 5, Section 5.1
(equivalent to assuming a Euclidean metric and identifying scalars with pseudo-scalars). It is to
obviate this inconsistency and restriction on generality that the approach adopted here bases its
definition of the interior product explicitly on the regressive exterior product.
Implications of the regressive product axioms
By expressing one or more elements as a complement, the relations of the regressive product
axiom set may be rewritten in terms of the interior product, thus yielding some of its more
fundamental properties.
m
���� ei
m
� ���� 6: The interior product of an m-element and a k-element is an (mÐk)-element.
The grade of the interior product of two elements is the difference of their grades.
Α m �� m , Β k
�� k � Α m ���� Β k
� �
m�k
Thus, in contradistinction to the regressive product, the grade of an interior product does not
depend on the dimension of the underlying linear space.
If the grade of the first factor is less than that of the second, the interior product is zero.
2001 4 5
Α m ���� Β k
� 0 m � k
6.3
6.4