Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.2. BASIC CONCEPTS OF GA 45
Corollary 2.9
The inner product of two blades A 〈k〉 and B 〈l〉 can be expressed by means of the
commutator formalism, that is
�
... × − (ak−2× − (ak−1× − (ak × − B 〈l〉 )))..., l even
A 〈k〉 · B 〈l〉 =
... × − (ak−2× − (ak−1× − (ak × − B 〈l〉 )))..., l odd.
The commutator expressions can be further expanded with the help of equation
(A.11) and equation (A.12): if the grade l of B 〈l〉 is even (odd), the former (latter)
equation has to be used.
Example 2.5 ( Expanded inner product ):
The inner product (a1 ∧ a2 ∧ a3) · B 〈5〉 may be expanded as
a1× − (a2× −(a3× − B 〈5〉 ))
(A.12)
= + a1a2a3B 〈5〉 + a1a2B 〈5〉 a3 − a1a3B 〈5〉 a2
− a1B 〈5〉 a3a2 + a2a3B 〈5〉 a1 + a2B 〈5〉 a3a1
− a3B 〈5〉 a2a1 − B 〈5〉 a3a2a1.
Note that changing the sign of the underlined summands (with an odd number of
trailing vectors) yields the outer product, see equation (A.11).
Proposition 2.5
Let A and B denote two general multivectors of �p,q. It then holds that
〈AB〉 = 〈BA〉.
Proof: Let A = �n i=0 〈A〉 i and B = �n j=0 〈B〉 j . It follows that
�
n�
�
〈AB〉 = 〈A〉 i 〈B〉 j =
i,j=0
n�
i,j=0
and further by the definition 2.3 of the inner product
〈AB〉 =
n�
i,j=0
δij
� �
〈A〉 i 〈B〉 j
= 〈A〉〈B〉 +
� �
〈A〉 i 〈B〉 j
According to corollary 2.6, it is 〈A〉 i ·〈B〉 i = 〈B〉 i ·〈A〉 i . Hence
〈AB〉 = 〈B〉〈A〉 +
n�
〈A〉 i · 〈B〉 i .
i=1
n�
〈B〉 i · 〈A〉 i = ... = 〈BA〉.
i=1
�