Hypercomplex Mathematics

The set of hypercomplex numbers is defined as

$$q= q_0+\sum_{l=1}^{n}i_lq_l \quad q_l \in \Re.$$ (1)

Hypercomplex numbers define a n+1-dimensional complex space with i_l orthonormal to i_m, for l $\neq$ m. For all cases presented in this paper l will be limited to 3. Such numbers are quaternions, which can be represented as q=q₀+iq₁+jq₂+kq₃, where i,j,k are imaginary numbers that satisfying the following relations:  

$$ij=-ji=k,~~~\text{and}~~~i^2=j^2=k^2=-1.$$ (2)

 

 

1 i j k

1 1 i j k

i i -1 k -j

j j -k -1 i

k k j -i -1

The multiplication table for quaternion unit vectors is shown in Table 1.
With these definitions, quaternionic addition between two quaternions, q and p, can be defined as

$$\begin{eqnarray} q+p~=~(q_0+iq_1+jq_2+kq_3)+(p_0+ip_1+jp_2+kp_3)\nonumber\ =~(q_0+p_0)+i(q_1+p_1)+j(q_2+p_2)+k(q_3+p_3),\end{eqnarray}$$ (3)

and multiplication as

$$\begin{eqnarray} qp~=~(q_0+iq_1+jq_2+kq_3)(p_0+ip_1+jp_2+kp_3)\nonumber\ =~(q_0... ...\nonumber\ +~k(q_0p_3+q_3p_0+q_1p_2-q_2p_1).~~~~~~~~~~~~~~~~~~~~~\end{eqnarray}$$ (4)

Notice that multiplication in equation 4, is not commutative due to the quaternionic algebra rules defined in table 1. Quaternions are often separated into two parts, q₀ and $\textbf{q}=iq_1+jq_2+kq_3$, respectively called the scalar and vector part of the quaternion. Using this definition, the conjugate of q, $\bar{q}$, is  

$$\bar{q}=q_0-\textbf{q}=q_0-iq_1-jq_2-kq_3$$ (5)

and the norm of a quaternion is defined by

$$\mid\mid q \mid\mid =\sqrt{q\bar{q}}=\sqrt{{q_0}^2+{q_1}^2+{q_2}^2+{q_3}^2}.$$ (6)

It is useful to formulate a polar representation of the quaternion, as this will be the primary notation throughout this paper. For any complex number, $\textbf{z}=a+ib$, the argument or phase-angle is defined as atan2(b,a). If $\textbf{z}$ is written it the form $\textbf{z}=\mid r \mid e^{i\gamma}$, then $\gamma$ is the phase (argument) of $\textbf{z}$, denoted arg($\textbf{z}$)=$\gamma$.Quaternions contain three complex subfields and, correspondingly, three phases that are the projections onto the i, j, or k-complex plane,

$$\begin{eqnarray} \phi=\text{arg}_{\text{i}}(\textbf{z})=\text{atan2}(q_1,q_0)~\... ...)~\ \psi=\text{arg}_{\text{k}}(\textbf{z})=\text{atan2}(q_3,q_0).\end{eqnarray}$$ (7) (8) (9)

 

• Quaternionic Transform