Quaternion Group/Complex Matrices

Representation of Quaternion Group
Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\MM_\C} 2$:


 * $\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:

Also see

 * Quaternions Defined by Matrices where it is shown that these have the appropriate properties.

In Matrix Form of Quaternion it is shown that a general element $\mathbf x$ of $\mathbb H$ has the form:
 * $\mathbf x = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - bi \end{bmatrix}$