Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software Split-biquaternions Compare with split-complex numbers. Will relate Split-biquaternions {$x+y\omega$} where {$\omega^2=1$} and {$x,y\in\mathbb{H}$} Clifford algebra {$x+y\epsilon_1$} as per the recursion: {$\mathbb{H}\otimes(\mathbb{R}\oplus\mathbb{R})$} Clifford algebra {$Cl_{0,3}(\mathbb{R})$} generated by {$e_1\rightarrow e_1\otimes 1, e_2\rightarrow e_2\otimes 1, e_3\rightarrow e_1e_2\otimes\epsilon_1$} where {$e_1^2=e_2^2=e_3^2=-1$} and {$\epsilon_1^2=1$} In other words, generated by {$i,j,k\omega$}. Direct sum {$(a,b)\in\mathbb{H}\oplus\mathbb{H}$} Matrix algebra {$\begin{pmatrix} x & y\\ y & x \end{pmatrix}$} {$\begin{pmatrix} x+y & 0\\ 0 & x-y \end{pmatrix}$} {$\begin{pmatrix} \frac{a}{2} & \frac{a}{2} & 0 & 0 \\ \frac{a}{2} & \frac{a}{2} & 0 & 0 \\ 0 & 0 & \frac{b}{2} & \frac{-b}{2} \\ 0 & 0 & \frac{-b}{2} & \frac{b}{2} \end{pmatrix}$} Matrix algebra {$i(a_{11} + a_{12}i + a_{13}j + a_{14}k + a_{21}\omega + a_{22}i\omega + a_{23}j\omega + a_{24}k\omega) = -a_{12} + a_{11}i -a_{14}j + a_{13}k -a_{22}\omega + a_{21}i\omega - a_{24}j\omega + a_{23}k\omega$} {$j(a_{11} + a_{12}i + a_{13}j + a_{14}k + a_{21}\omega + a_{22}i\omega + a_{23}j\omega + a_{24}k\omega) = -a_{13} + a_{14}i +a_{11}j-a_{12}k - a_{23}\omega + a_{24}i\omega + a_{21}j\omega - a_{22}k\omega$} {$k\omega(a_{11} + a_{12}i + a_{13}j + a_{14}k + a_{21}\omega + a_{22}i\omega + a_{23}j\omega + a_{24}k\omega) = -a_{24}-a_{23}i+a_{22}j+a_{21}k -a_{14}\omega -a_{13}i\omega +a_{12}j\omega +a_{11}k\omega$}
This page was last changed on April 26, 2024, at 11:38 PM