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The Quaternion group, ##Q=\{1,-1,i,-i,j,-j,k,-k\}##, can be realized by ##2x2## matricies:

##

\begin{align*}

1=\begin{bmatrix} 1,0 \\ 0,1\end{bmatrix} &\hspace{10pt} i=\begin{bmatrix} \omega,0 \\ 0,-\omega\end{bmatrix} & \hspace{10pt}j=\begin{bmatrix} 0,1 \\ -1,0\end{bmatrix} & \hspace{10pt}k=\begin{bmatrix} 0,\omega \\ \omega,0\end{bmatrix}

\end{align*}

##

with ##\omega^2=-1##.

I was told ##Q## can also be represented (non-trivially)by ##3x3## or ##4x4## matricies but could not find any source explaining this and was hoping someone here could either provide a reference or explain this a bit.

Thanks,

Jack

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# Can quaternion group be represented by 3x3 matricies?

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