Grassmann Algebra

TheInteriorProduct.nb 3
The basing of the notion of interior product on the notions of regressive product and
complement follows here the Grassmannian tradition rather than that of the current literature
which introduces the inner product onto a linear space as an arbitrary extra definition. We do
this in the belief that it is the most straightforward way to obtain consistency within the algebra
and to see and exploit the relationships between the notions of exterior product, regressive
product, complement and interior product, and to discover and prove formulae relating them.
We use the term 'interior' in addition to 'inner' to signal that the products are not quite the same.
In traditional usage the inner product has resulted in a scalar. The interior product is however
more general, being able to operate on two elements of any and perhaps different grades. We
reserve the term inner product for the interior product of two elements of the same grade. An
inner product of two elements of grade 1 is called a scalar product. In sum: inner products are
scalar whilst, in general, interior products are not.
We denote the interior product with a small circle with a 'bar' through it, thus ����. This is to
signify that it has a more extended meaning than the inner product. Thus the interior product of
Α m with Β k becomes Α m ���� Β k . This product is zero if m < k. Thus the order is important: the element
of higher degree should be written on the left. The interior product has the same left
associativity as the negation operator, or minus sign. It is possible to define both left and right
interior products, but in practice the added complexity is not rewarded by an increase in utility.
We will see in Chapter 10: The Generalized Product that the interior product of two elements
can be expressed as a certain generalized product, independent of the order of the factors.
6.2 Defining the Interior Product
Definition of the interior product
The interior product of Α m and Β k
If m < k, then Α m � Β k
hence:
is denoted Α m ���� Β k
Α ���� Β �Α� Β
m k m k
����� � �
and is defined by:
m�k
m � k
����� is necessarily zero (otherwise the grade of the product would be negative),
An important convention
Α m ���� Β k
� 0 m � k
In order to avoid unnecessarily distracting caveats on every formula involving interior products,
in the rest of this book we will suppose that the grade of the first factor is always greater than
or equal to the grade of the second factor. The formulae will remain true even if this is not the
case, but they will be trivially so by virtue of their terms reducing to zero.
2001 4 5
6.1
6.2