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:
- $\begin{array}{r|rrrrrrrr}
& \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\
\hline
\mathbf 1 & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \mathbf i & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf k & -\mathbf j & -\mathbf k & \mathbf j \\
-\mathbf 1 & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf j & -\mathbf k & \mathbf j & \mathbf k \\ -\mathbf i & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf k & \mathbf j & \mathbf k & -\mathbf j \\
\mathbf j & \mathbf j & -\mathbf k & -\mathbf j & \mathbf k & -\mathbf 1 & \mathbf i & \mathbf 1 & -\mathbf i \\ \mathbf k & \mathbf k & \mathbf j & -\mathbf k & -\mathbf j & -\mathbf i & -\mathbf 1 & \mathbf i & \mathbf 1 \\
-\mathbf j & -\mathbf j & \mathbf k & \mathbf j & -\mathbf k & \mathbf 1 & -\mathbf i & -\mathbf 1 & \mathbf i \\ -\mathbf k & -\mathbf k & -\mathbf j & \mathbf k & \mathbf j & \mathbf i & \mathbf 1 & -\mathbf i & -\mathbf 1 \end{array}$
Also see
- 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:
- $\mathbf x = \begin {bmatrix} a + b i & c + d i \\ -c + d i & a - bi \end {bmatrix}$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(6) \ \text{(ii)}$