Conformal Geometric Algebra in Stochastic Optimization Problems ...

2.3. EXTENDED CONCEPTS OF GA 73
Z ′ 〈k〉 = � k
i=1 z′ i , respectively. It follows n ∧Z 〈k〉 = n ∧Z′ 〈k〉 and even Z 2 〈k〉 = Z′2 〈k〉.
But Z ′ 〈k〉 has a certain offset n ∧ O 〈k−1〉
Z ′ 〈k〉 =
k�
i=1
where O 〈k−1〉 =
(zi + λin) = Z 〈k〉 + n ∧ O 〈k−1〉 ,
k�
(−1) j−1 λj [Z 〈k〉 \zj].
j=1
It is therefore not desirable to work in the primed basis � ′ . In order to fully remove
the ambiguousness the following operation, which will later on be called rejection,
can be used
z := z ′ − n† · z ′
n † · n n, z ∈ {z′ 1...k }.
As a result it is obtained that n † ·z = n ∗ε z = 0 in addition to n ·z = n † ∗ε z = 0.
The definition generalizes this concept.
Definition 2.20 ( Sound basis ):
Let � ′ = {n ′ 1...r } ∪ {z′ 1...s } denote an orthogonal basis such that additionally
n ′2
i = 0, i ∈ [1,r] � . Then the sound basis is defined as � = {n1...r } ∪ {z1...s }, such
that ∀x, y ∈ � : x �= y ⇔ x · y = 0, and moreover, ∀x ∈ �, ∀y ∈ {n1...r } : x �= y
⇔ x ∗ε y = 0.
�
Hence all elements of a sound basis are mutually orthogonal. In addition, the null
vectors are mutually perpendicular. Finally, the space span{z 1...s } is perpendicular
to the space span{n 1...r }. Recalling that a ∗ε b = a † · b demonstrates that the
conjugate of each vector in {n 1...r } is orthogonal to the remaining ones.
In order to achieve that the null vectors become mutually perpendicular as well,
a Gram-Schmidt orthogonalization of {n ′ 1...r }, carried out in the Euclidean space
� n , can be used. The i th vector ni can then be represented by the linear transformation
ni = n ′ �
i − j