Dihedral Group D4/Matrix Representation/Formulation 1
Matrix Representation of Dihedral Group $D_4$
Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C$ denote the following four elements of the matrix space $\map {\MM_\Z} 2$:
- $\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
The set:
- $D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$
under the operation of conventional matrix multiplication, forms the dihedral group $D_4$.
Cayley Table
Its Cayley table is given by:
- $\begin{array}{r|rrrrrrrr}
& \mathbf I & \mathbf A & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\
\hline
\mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C \\ \mathbf A & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B \\ \mathbf B & \mathbf B & \mathbf C & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A \\ \mathbf C & \mathbf C & \mathbf B & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I \\
-\mathbf I & -\mathbf I & -\mathbf A & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf C \\ -\mathbf A & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I & -\mathbf C & -\mathbf B \\ -\mathbf B & -\mathbf B & -\mathbf C & \mathbf I & \mathbf A & \mathbf B & -\mathbf C & -\mathbf I & -\mathbf A \\ -\mathbf C & -\mathbf C & -\mathbf B & -\mathbf A & -\mathbf I & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$
Also see
- Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties.
Sources
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