Conformal Geometric Algebra in Stochastic Optimization Problems ...

60 CHAPTER 2. GEOMETRIC ALGEBRA
Proposition 2.11
Let e � and e � two basis blades of �p,q. Then the conjugate (e �e �) † of the geometric
product e �e � is
(e �e �) † = e † �e † �.
Proof: By means of equation (2.35) a scalar x can be determined such that e �e � =
(−1) x e �e �. However, exchanging the basis blades twice, as in the last line, introduces
no sign.
(e �e �) † = e † �e † �
(e �e �) 2 (e �e �) = e 2 � e 2 � e �e �
e �e �
� �� �
swap
e �e � e �e �
� �� �
swap
= e �e � e �e � e �e �
By the distributivity of the geometric product, and since inner and outer product
are also defined in terms of the geometric product, these rules apply
and especially
(AB) † = B † A †
(A ∧ B) † = B † ∧ A †
A †
〈k〉
(A · B) † = B † · A †
= a†
k ∧ a†
k−1 ∧ ... ∧ a† 1 .
In this respect, it is important that generally
a † �= ±a and thus A †
〈k〉 �= ±A 〈k〉
because the number of basis vectors with negative signature may vary between the
basis blades of A 〈k〉 . Let x = e1 + e2 ∈ � 1,1 . Then, the result of x ∧ x † = −2e12
shows that for general x ∈ � p,q
x ∧ A 〈k〉 = 0 �=⇒ x ∧ A †
〈k〉 = 0.
Note that the conjugate and reverse operation commute as their result ultimately
differs by a sign.
By means of the conjugate the signature independent Euclidean inner product and
the likewise signature independent Euclidean scalar product may be defined.
Definition 2.11 (Euclidean inner product ):
Given two basis blades e� and e� of �p,q their Euclidean inner product, denoted
by e� ·ε e�, is defined by
�
†
e � · e�, |�| ≤ |�|
e � ·ε e � =
e � · e † �, else.
�
�