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¯
·¯
¯
ׯ
Let
p
q
and
p
q
denote the dot and cross products between the three dimensional
¯
¯
vectors
p
q
. Then basic operations between quaternions can be defined as follows:
•
The addition and subtraction of quaternions are defined in a similar manner as for
complex-valued numbers:
p
±
q
=
(
p
r
±
q
r
,
¯
p
±¯
q
)
=
(
p
r
±
q
r
,
p
i
±
q
i
,
p
j
±
q
j
,
p
k
±
q
k
)
(2.9)
•
The product of p and q is determined using Eq.
2.8
as
pq
=
(
p
r
q
r
−¯
p
·¯
q
,
p
r
q
¯
+
q
r
p
¯
+¯
p
ׯ
q
)
(2.10)
•
The conjugate of the product is defined as
)
ℵ
=
q
ℵ
p
ℵ
(
pq
(2.11)
•
The quaternion norm of
q
, denoted by
|
q
|
, is defined as
qq
ℵ
=
q
r
q
i
q
j
+
q
k
|
q
| =
+
+
(2.12)
2.5.1.2 Cauchy-Riemann-Fueter Equation
The Swiss mathematician Fueter developed the appropriate generalization of the
Cauchy-Riemann equations to the quaternionic functions. The analytic condition
for the quaternionic functions is defined by Cauchy-Riemann-Fueter (CRF) equa-
tion, which corresponds as an extension of the Cauchy-Riemann (CR) equations
defined for the functions in complex domain. In order to construct learning rules for
quaternionic neural networks, CRF equation describes the required analyticity (or
differentiability) of the function in the quaternionic domain.
Definition 2.12
H
be a quaternionic valued function defined over a
quaternionic variable. The condition for differentiability of any quaternionic function
f
is defined as follows:
Let
f
:
H
→
∂
f
(
q
)
i
∂
f
(
q
)
j
∂
f
(
q
)
k
∂
f
(
q
)
=−
=−
=−
(2.13)
∂
q
r
∂
q
i
∂
q
j
∂
q
k
An analytic function can serve as the activation function in the neural network. The
analytic condition for the quaternionic function, called the Cauchy-Riemann-Fueter
(CRF) equation, yields:
∂
f
(
q
)
i
∂
f
(
q
)
j
∂
f
(
q
)
k
∂
f
(
q
)
+
+
+
=
0
(2.14)
∂
∂
∂
∂
q
r
q
i
q
j
q
k
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