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} $ $=\begin{bmatrix} \begin{bmatrix} \alpha_{00}\omega_{00}-\omega_{01}\alpha_{01}-\omega_{10}\alpha_{10}-\alpha_{11}\omega_{11}\\ \alpha_{01}\omega_{00}+\omega_{01}\alpha_{00}-\omega_{10}\alpha_{11}+\alpha_{10}\omega_{11} \end{bmatrix} \\ \begin{bmatrix} \alpha_{10}\omega_{00}+\omega_{01}\alpha_{11}+\omega_{10}\alpha_{00}-\alpha_{01}\omega_{11}\\ \alpha_{11}\omega_{00}-\omega_{01}\alpha_{10}+\omega_{10}\alpha_{01}+\alpha_{00}\omega_{11} \end{bmatrix} \end{bmatrix}$

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} $ $=\begin{bmatrix} \begin{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix} \begin{bmatrix}\omega_{000}\\\omega_{001}\end{bmatrix} + \begin{bmatrix}\overline{\omega_{010}}\\\omega_{011}\end{bmatrix} \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} + \begin{bmatrix}\overline{\omega_{100}}\\\omega_{101}\end{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix} + \begin{bmatrix}\overline{\alpha_{110}}\\\alpha_{111}\end{bmatrix} \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \\ \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} \begin{bmatrix}\omega_{000}\\\overline{\omega_{001}}\end{bmatrix} + \begin{bmatrix}\omega_{010}\\\omega_{011}\end{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix} + \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \begin{bmatrix}\alpha_{100}\\\overline{\alpha_{101}}\end{bmatrix} + \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} \begin{bmatrix}\overline{\omega_{100}}\\\omega_{101}\end{bmatrix} \end{bmatrix}\\ \begin{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix} \begin{bmatrix}\omega_{000}\\\overline{\omega_{001}}\end{bmatrix} + \begin{bmatrix}\omega_{010}\\\overline{\omega_{011}}\end{bmatrix} \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} + \begin{bmatrix}\omega_{100}\\\omega_{101}\end{bmatrix} \begin{bmatrix}\alpha_{000}\\\alpha_{001}\end{bmatrix} + \begin{bmatrix}\overline{\alpha_{010}}\\\alpha_{011}\end{bmatrix} \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \\ \begin{bmatrix}\alpha_{110}\\\alpha_{111}\end{bmatrix} \begin{bmatrix}\omega_{000}\\\omega_{001}\end{bmatrix} + \begin{bmatrix}\overline{\omega_{010}}\\\overline{\omega_{011}}\end{bmatrix} \begin{bmatrix}\alpha_{100}\\\alpha_{101}\end{bmatrix} + \begin{bmatrix}\omega_{110}\\\omega_{111}\end{bmatrix} \begin{bmatrix}\alpha_{000}\\\overline{\alpha_{001}}\end{bmatrix} + \begin{bmatrix}\alpha_{010}\\\alpha_{011}\end{bmatrix} \begin{bmatrix}\omega_{100}\\\omega_{101}\end{bmatrix} \end{bmatrix} \end{bmatrix}$ $=\begin{bmatrix}\begin{bmatrix}\begin{bmatrix}

\alpha_{000}\omega_{000}-\omega_{001}\alpha_{001}

-\omega_{010}\alpha_{010}-\alpha_{011}\omega_{011} -\omega_{100}\alpha_{100}-\alpha_{101}\omega_{101} -\alpha_{110}\omega_{110}-\omega_{111}\alpha_{111} \\

\alpha_{001}\omega_{000}+\omega_{001}\alpha_{000}

+\omega_{011}\alpha_{010}-\alpha_{011}\omega_{010} +\omega_{101}\alpha_{100}-\alpha_{101}\omega_{100} +\alpha_{111}\omega_{110}-\omega_{111}\alpha_{110} \end{bmatrix}\\\begin{bmatrix}

\alpha_{010}\omega_{000}+\omega_{001}\alpha_{011}

-\omega_{010}\alpha_{000}-\alpha_{001}\omega_{011} +\omega_{110}\alpha_{100}+\alpha_{101}\omega_{111} -\alpha_{110}\omega_{100}-\omega_{101}\alpha_{111} \\

\alpha_{011}\omega_{000}-\omega_{001}\alpha_{010}

+\omega_{011}\alpha_{000}+\alpha_{001}\omega_{010} +\omega_{111}\alpha_{100}-\alpha_{101}\omega_{110} -\alpha_{111}\omega_{100}+\omega_{101}\alpha_{110} \end{bmatrix}\end{bmatrix}\\\begin{bmatrix}\begin{bmatrix}

\alpha_{100}\omega_{000}+\omega_{001}\alpha_{101}

-\omega_{010}\alpha_{110}-\alpha_{111}\omega_{011} +\omega_{100}\alpha_{000}-\alpha_{001}\omega_{101} -\alpha_{010}\omega_{110}-\omega_{101}\alpha_{111} \\

\alpha_{101}\omega_{000}-\omega_{001}\alpha_{100}

-\omega_{011}\alpha_{110}+\alpha_{111}\omega_{010} +\omega_{101}\alpha_{000}+\alpha_{001}\omega_{100} +\alpha_{011}\omega_{110}-\omega_{111}\alpha_{010} \end{bmatrix}\\\begin{bmatrix}

\alpha_{110}\omega_{000}-\omega_{001}\alpha_{111}

-\omega_{010}\alpha_{100}+\alpha_{101}\omega_{011} +\omega_{110}\alpha_{000}+\alpha_{001}\omega_{111} +\alpha_{010}\omega_{100}-\omega_{101}\alpha_{011} \\

\alpha_{111}\omega_{000}+\omega_{001}\alpha_{110}

-\omega_{011}\alpha_{100}-\alpha_{101}\omega_{010} +\omega_{111}\alpha_{000}-\alpha_{001}\omega_{110} +\alpha_{011}\omega_{100}+\omega_{101}\alpha_{010} \end{bmatrix}\end{bmatrix}\end{bmatrix}$