User:Roman Czyborra/1st

Dickson Multiplication in General
$\begin{bmatrix}\alpha_0\\\alpha_1\end{bmatrix}\begin{bmatrix}\omega_0\\\omega_1\end{bmatrix} $ := $\begin{bmatrix}\alpha_0\omega_0+\overline{\omega_1}\alpha_1\\ \alpha_1\underline{\omega_0}+\omega_1\alpha_0\end{bmatrix}$ where $\underline x:=x^*=:-\overline x$.

Dickson Multiplication in $\R^1=\R$
$\begin{bmatrix}\alpha_0\\0\end{bmatrix}\begin{bmatrix}\omega_0\\0\end{bmatrix} $ $:=\begin{bmatrix}\alpha_0\omega_0+\overline{0}0\\0\underline{\omega_0}+0\alpha_0\end{bmatrix} $ $=\begin{bmatrix}\alpha_0\omega_0+0\\0+0\end{bmatrix} $ $=\begin{bmatrix}\alpha_0\omega_0\\0\end{bmatrix}$.

Dickson Multiplication in $\R^2=\C$
$\begin{bmatrix}\alpha_0\\\alpha_1\end{bmatrix}\begin{bmatrix}\omega_0\\\omega_1\end{bmatrix} $ $:=\begin{bmatrix}\alpha_0\omega_0+\overline{\omega_1}\alpha_1\\ \alpha_1\underline{\omega_0}+\omega_1\alpha_0\end{bmatrix} $ $=\begin{bmatrix}\alpha_0\omega_0-\omega_1\alpha_1\\ \alpha_1\omega_0+\omega_1\alpha_0\end{bmatrix} $

Dickson Multiplication in $\R^4=\mathbb H$
$ \begin{bmatrix} \begin{bmatrix}\alpha_{00}\\\alpha_{01}\end{bmatrix}\\ \begin{bmatrix}\alpha_{10}\\\alpha_{11}\end{bmatrix} \end{bmatrix} \begin{bmatrix} \begin{bmatrix}\omega_{00}\\\omega_{01}\end{bmatrix}\\ \begin{bmatrix}\omega_{10}\\\omega_{11}\end{bmatrix} \end{bmatrix} $ $=\begin{bmatrix} \begin{bmatrix}\alpha_{00}\\\alpha_{01}\end{bmatrix} \begin{bmatrix}\omega_{00}\\\omega_{01}\end{bmatrix} + \begin{bmatrix}\overline{\omega_{10}}\\\omega_{11}\end{bmatrix} \begin{bmatrix}\alpha_{10}\\\alpha_{11}\end{bmatrix} \\ \begin{bmatrix}\alpha_{10}\\\alpha_{11}\end{bmatrix} \begin{bmatrix}\omega_{00}\\\overline{\omega_{01}}\end{bmatrix} + \begin{bmatrix}\omega_{10}\\\omega_{11}\end{bmatrix} \begin{bmatrix}\alpha_{00}\\\alpha_{01}\end{bmatrix} \end{bmatrix} $ $=\begin{bmatrix} \begin{bmatrix} \alpha_{00}\omega_{00}-\omega_{01}\alpha_{01}-\omega_{10}\alpha_{10}-\omega_{11}\alpha_{11}\\ \alpha_{01}\omega_{00}+\omega_{01}\alpha_{00}-\omega_{10}\alpha_{11}+\omega_{11}\alpha_{10} \end{bmatrix} \\ \begin{bmatrix} \alpha_{10}\omega_{00}+\omega_{01}\alpha_{11}+\omega_{10}\alpha_{00}-\omega_{11}\alpha_{01}\\ \alpha_{11}\omega_{00}-\omega_{01}\alpha_{10}+\omega_{10}\alpha_{01}+\omega_{11}\alpha_{00} \end{bmatrix} \end{bmatrix}$

Dickson Multiplication in $\R^8=\mathbb O$
$\begin{bmatrix}\begin{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix}\\ \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} \end{bmatrix}\\\begin{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix}\\ \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} \end{bmatrix}\end{bmatrix} \begin{bmatrix}\begin{bmatrix} \begin{bmatrix}\omega_{000}\\\omega_{001}\end{bmatrix}\\ \begin{bmatrix}\omega_{010}\\\omega_{011}\end{bmatrix} \end{bmatrix}\\\begin{bmatrix} \begin{bmatrix}\omega_{100}\\\omega_{101}\end{bmatrix}\\ \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \end{bmatrix}\end{bmatrix} $ = $\begin{bmatrix} \begin{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix}\\ \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} \end{bmatrix}\begin{bmatrix} \begin{bmatrix}\omega_{000}\\\omega_{001}\end{bmatrix}\\ \begin{bmatrix}\omega_{010}\\\omega_{011}\end{bmatrix} \end{bmatrix} + \begin{bmatrix} \begin{bmatrix}\overline{\omega_{100}}\\\omega_{101}\end{bmatrix}\\ \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \end{bmatrix}\begin{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix}\\ \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} \end{bmatrix} \\ \begin{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix}\\ \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} \end{bmatrix}\begin{bmatrix} \begin{bmatrix}\omega_{000}\\\overline{\omega_{001}}\end{bmatrix}\\ \begin{bmatrix}\overline{\omega_{010}}\\\overline{\omega_{011}}\end{bmatrix} \end{bmatrix} + \begin{bmatrix} \begin{bmatrix}\omega_{100}\\\omega_{101}\end{bmatrix}\\ \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \end{bmatrix}\begin{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix}\\ \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} \end{bmatrix} \end{bmatrix} $