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2.5.1.1 Beauty of Quaternionic Numbers
A quaternion, the generalization of complex number, is a hypercomplex number
where complex analysis would be self evident within the structure of quaternion
analysis. Unlike the complex number, the quaternion has four components: one is
real and the other three are all imaginary.
Definition 2.7
A class of hypercomplex numbers, the quaternions, are defined as a
vector
q
in a four-dimensional vector space over the real numbers (
R
) with an ordered
basis. Each number is a quadruple consisting a real number and three imaginary
numbers i, j, and k. A quaternion
q
∈
H
is expressed by fundamental formula
q
=
q
r
+
q
i
i
+
q
j
j
+
q
k
k
(2.4)
where
q
r
,
q
j
and
q
k
are real numbers. The set of quaternions
H
, which is equal
to
R
4
, constitutes a four dimensional vector space over the real numbers with basis
{
q
i
,
1
;
i
;
j
;
k
}
.
Definition 2.8
The quaternion
q
∈
H
can also be interpreted as having a real part
q
r
and vector part
q
, where the elements
¯
{
i
,
j
,
k
}
are given an added geometric
interpretation as unit vectors along the
X
Z
axis respectively. Equation
2.4
can
also be written using 4-tuple or 2-tuple (one scalar and one vector in three space)
notation as
,
Y
,
=
(
q
r
,
q
i
,
q
j
,
q
k
)
=
(
q
r
,
¯
)
q
q
(2.5)
where
0
k
of
quaternions may be regarded as being equivalent to the real numbers. The subspace
q
q
¯
=
q
i
,
q
j
,
q
k
. Accordingly, the subspace
q
=
q
r
+
0
i
+
0
j
+
=
0
+
q
i
i
+
q
j
j
+
q
k
k
may be regarded as being equivalent to the ordinary
3D vector in
R
3
.
Definition 2.9
In Wirtinger calculus, a quaternion and its conjugate are treated as
independent of each other, which makes the derivation of the learning scheme in
neural network clear and compact. The quaternion conjugate is defined as
q
ℵ
=
(
q
r
,
−¯
q
)
=
q
r
−
q
i
i
−
q
j
j
−
q
k
k
(2.6)
Definition 2.10
According to the Hamilton rule the quaternion basis satisfy the
following identities, which immediately follows that multiplication of quaternions
is not commutative.
i
2
j
2
k
2
=
=
=
ijk
=−
1
(2.7)
ij
=−
ji
=
k
;
jk
=−
kj
=
i
;
ki
=−
ik
=
j
;
(2.8)
Definition 2.11
. Customarily, the extension of
an algebra attempts to preserve the basic operations defined in the original algebra.
Let
p
=
(
p
r
,
¯
p
)
and
q
=
(
q
r
,
¯
q
)
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