multiplication table

The best way to get at 2x2 matrices, is to use quaternions. Starting from the natural basis for 2x2 matrices,

$$\begin{array}{cccc} \begin{array}{rcl} {\bf e}_{11} & = & ... ...0 & 0 \\ 0 & 1 \end{array} \right]. \end{array} \end{array}$$

whose rule of multiplication is ${\bf e}_{ij} {\bf e}_{kl} = \delta_{jk} {\bf e}_{il}$, quaternion-like matrices can be defined by

$$\begin{array}{cccc} \begin{array}{rcl} {\bf 1}& = & \left... ... & 0 \\ 0 & -1 \end{array} \right]. \end{array} \end{array}$$

In detail,

$${\bf 1}= {\bf e}_{11} + {\bf e}_{22},\ \ {\bf i}= {\bf e}_{12... ...}_{12} + {\bf e}_{21},\ \ {\bf k}= {\bf e}_{11} - {\bf e}_{22},$$

all built from sums and differences, thereby retaining real matrices. Like quaternions, these matrices anticommute (except for the identity). The ostensible difference is that only one square is $-{\bf 1}$, the others are $+{\bf 1}$, changing Euler's formula from trigonometric to hyperbolic functions according to the sign.

The multiplication table is

$$\begin{array}{r\vert rrrr} & {\bf 1}& {\bf i}& {\bf j}& {\b... ...}\\ {\bf k}& {\bf k}& {\bf j}& {\bf i}& {\bf 1} \end{array}.$$

The usual way of performing algebraic operations on these matrices is to write a sum such as $a {\bf 1}+ b {\bf i}+ c {\bf j}+ d {\bf k}$ in the form $s + {\bf v}$, where $s = a {\bf 1}$ and ${\bf v}$ is the rest of the sum. One can adopt the custom of not writing an explicit ${\bf 1}$ in places where the meaning is clear.