Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.3. EXTENDED CONCEPTS OF GA 71
where each element c ∈ �er(A 〈k〉 ∧ B 〈l〉 ) corresponds to a uniquely defined pair
(a, b) ∈ �er(A 〈k〉 ) × �er(B 〈l〉 ) such that c = a + b. Regarding the outer sum of the
vector spaces � and �, as above, {c 1...t } = {} is required.
From section 2.3.3, and especially from equation (2.54), it can be deduced that
�er(A 〈k〉 ) = �er ∗ (A ∗ 〈k〉 ) �er(A∗ 〈k〉 ) = �er∗ (A 〈k〉 )
�er(A 〈k〉 ) ⊕ �er ∗ (A 〈k〉 ) = �er(A 〈k〉 ) ⊕ �er(A ∗ 〈k〉 ) = �p,q ,
where in case of a null blade A 〈k〉 , the outer sum is not a direct sum any more
because it exists a common subspace, i.e. dim( �er(A 〈k〉 ) ∩ �er(A∗ 〈k〉 )) is greater
than zero. This is detailed in the section starting at page 72.
Definition 2.19 ( Inner difference ):
Let �, � ⊆ � p,q . Then their (inner) difference, denoted by � ⊖ �, is defined as
Especially, let
denote the Euclidean inner difference.
� ⊖ � = {a ∈ � | ∀b ∈ � : a · b = 0 }.
� ⊖ε � = {a ∈ � | ∀b ∈ � : a ∗ε b = 0 }
Fig. 2.7: Orthogonal complements: the spaces indicated by the blue and the red
line are orthogonal. Similarly, a � and a⊥ are assumed to be orthogonal. The blue
‘decomposition’ of a is meant to be perpendicular.
Just as the outer product is related to the outer sum, the inner product is related
to the inner difference: by the considerations on page 51, it is known that a · B 〈l〉
is the orthogonal complement of a in B 〈l〉 if B2 〈l〉 �= 0. This exactly matches the
definition of the inner difference if a · B 〈l〉 �= 0
Hence if B 2 〈l〉 �= 0 and a · B 〈l〉
�er(a · B 〈l〉 ) = �er(B 〈l〉 ) ⊖ �er(a).
�= 0, it may be written
�
�
�er((a · B 〈l〉 ) · B 〈l〉 ) = �er(B 〈l〉 ) ⊖ �er(B 〈l〉 ) ⊖ �er(a) , l ≥ 2.
�