Generator for Quaternion Group

Theorem
The Quaternion Group can be generated by the matrices:


 * $\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}

\qquad \mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

where $i$ is the imaginary unit:
 * $i^2 = -1$

Proof
Note that:
 * $\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

is the identity for (conventional) matrix multiplication of order $2$.

We have:

and so:

Next we have:

and so:
 * $\mathbf b^2 = \mathbf a^2$

Then we have:

Thus $\gen {\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} }$ fulfils the conditions for the group presentation of $\Dic 2$:

Hence the result.