Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.2. BASIC CONCEPTS OF GA 53
Example 2.8 ( Basis orthogonalization ):
Here the orthogonal complement of [A 〈k〉 \ai] in A 〈k〉 is to be analyzed. Let A 〈k〉 be
a non-null blade, i.e. A2 〈k〉 �= 0, with basis {a1...k }. Then
�
[A 〈k〉 \ai] A 〈k〉 = [A 〈k〉 \ai] [A 〈k〉 \ai] ∧ (−1) k−i �
ai
Hence with
it follows ∀i,j ∈ [1,k] � :
= (−1) k−i [A 〈k〉 \ai] 2 ai⊥.
bi := (−1) k−i [A 〈k〉 \ai] A 〈k〉
[A 〈k〉 \ai] 2 ,
i �= j ⇐⇒ bi · aj = 0
Starting from the expansion of a · A [k] , given in equation (2.23), it is now being
focused on the expansion of a · A 〈k〉 : proposition 2.7 is of fundamental significance
as it, for example, expresses a kind of distributive law for the inner product over
the outer product.
Proposition 2.7
Given a vector a and a blade B 〈l〉 = � l
i=1 bi the following expansion can be used
a · B 〈l〉 =
l�
(−1) i−1 (a · bi) [B 〈l〉 \bi]
i=1
Proof: Let B = b1b2 ...bl the geometric product of the vectors {b1...l } such that
a · B 〈l〉 = a · 〈B〉 l = 〈a · B〉 l−1 . Consequently, it is
a · B 〈l〉 = 1
�
2 a 〈B〉 l − (−1) l �
〈B〉 la � � l
= aB − (−1) Ba �
� 1
2
According to the derivations on page 30, the inner term can be replaced as
�
l�
�
a · B 〈l〉
=
=
=
i=1
l−1 .
(−1) i−1 (a · bi)b1b2 ... ˇ bi ...bl
l�
(−1) i−1 (a · bi) 〈b1b2 ... ˇ bi ...bl〉 l−1
i=1
l−1
l�
(−1) i−1 (a · bi) 〈b1b2 ... ˇ bi ...bl〉 l−1
� �� �
[B 〈l〉 \bi]
i=1
�