Quaternion Group/Complex Matrices

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Representation of Quaternion Group

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\mathcal M_\C} 2$:

$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

where $\C$ is the set of complex numbers.


The set:

$\Dic 2 = \set {\mathbf 1, -\mathbf 1, \mathbf i, -\mathbf i, \mathbf j, -\mathbf j, \mathbf k, -\mathbf k}$

under the operation of conventional matrix multiplication, forms the quaternion group:


Cayley Table

Its Cayley table is given by:

$\begin{array}{r|rrrrrrrr} & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \hline \mathbf 1 & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \mathbf i & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf k & -\mathbf j & -\mathbf k & \mathbf j \\ -\mathbf 1 & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf j & -\mathbf k & \mathbf j & \mathbf k \\ -\mathbf i & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf k & \mathbf j & \mathbf k & -\mathbf j \\ \mathbf j & \mathbf j & -\mathbf k & -\mathbf j & \mathbf k & -\mathbf 1 & \mathbf i & \mathbf 1 & -\mathbf i \\ \mathbf k & \mathbf k & \mathbf j & -\mathbf k & -\mathbf j & -\mathbf i & -\mathbf 1 & \mathbf i & \mathbf 1 \\ -\mathbf j & -\mathbf j & \mathbf k & \mathbf j & -\mathbf k & \mathbf 1 & -\mathbf i & -\mathbf 1 & \mathbf i \\ -\mathbf k & -\mathbf k & -\mathbf j & \mathbf k & \mathbf j & \mathbf i & \mathbf 1 & -\mathbf i & -\mathbf 1 \end{array}$


Also see

In Matrix Form of Quaternion it is shown that a general element $\mathbf x$ of $\mathbb H$ has the form:

$\mathbf x = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - bi \end{bmatrix}$


Sources