Conformal Geometric Algebra in Stochastic Optimization Problems ...

64 CHAPTER 2. GEOMETRIC ALGEBRA
where the matrix A holds the coefficients of the vectors {a 1...n }. As the pseudoscalar
I is proportional to the outer product of n linearly independent vectors,
the pseudoscalar spans the whole space � p,q . This has profound ramifications: the
geometric product of a multivector A and I, and so I −1 , is always identical to the
inner product of the two elements because I comprises all n basis vectors such that
equation (2.37) applies. Thus for every A ∈ �p,q
A ∗ = AI −1 = A · I −1 .
Recall from corollary 2.8 that (A [r] ∧ B [s] ) · C [t] = A [r] · (B [s] · C [t] ) iff t ≥ r + s,
which now reads
or simply
(A [r] ∧ B [s] )I −1 = A [r] · (B [s] I −1 ), if n ≥ r + s,
(A [r] ∧ B [s] ) ∗ = A [r] · B ∗ [s] . (2.50)
(A [r] · B [s] ) ∗ = A [r] ∧ B ∗ [s] , s ≥ r, (2.51)
The second very similar identity can be derived from equation (2.50) by
A [r] ∧ B ∗ [s] = A [r] ∧ (B [s] I −1 ��
) = A [r] ∧ (B [s] I −1 � �
) I I −1
(2.50)
=
�
A [r] · (B [s] I −1 �
I) I −1
= (A [r] · B [s] ) ∗ .
The second line follows if n ≥ r + (n − s) hence if s ≥ r.
In case of a vector and a general multivector the following rules apply:
(a ∧ B) ∗ = a · B ∗
(2.52)
(a · B) ∗ = a ∧ B ∗ . (2.53)
Again, the second identity arises from the first one 12 via
a ∧ B ∗
= a ∧ (BI −1 ) = �� a ∧ (BI −1 ) � I � I −1
(2.52)
= � a · (BI −1 I) � I −1
= (a · B) ∗ .
With the help of the dual, a relation between corollary 2.7 and corollary 2.13 can
be established. Given a vector x and a blade A 〈k〉 it is known from the corollaries
that
x ∧ A 〈k〉 = 0 iff x ∈ A 〈k〉 and x · A 〈k〉 = 0 iff x ⊥ A 〈k〉 .
12 Note that the first identity (a ∧ B[l]) ∗ = a · B ∗
[l] also holds for l = n as both sides take on the
value zero.