Complex Matrix

A Matrix whose elements may contain Complex Numbers. The Matrix Product of two $2\times 2$ complex matrices is given by

$\left[{\matrix{x_{11}+y_{11}i & x_{12}+y_{12}i\cr x_{21}+y_{21}i & x_{22}+y_{22... ...{u_{11}+v_{11}i & u_{12}+v_{12}i\cr u_{21}+v_{21}i & u_{22}+v_{22}i\cr}}\right]$
$=\left[{\matrix{R_{11} & R_{12}\cr R_{21} & R_{22}}}\right] + i\left[{\matrix{I_{11} & I_{12}\cr I_{21} & I_{22}}}\right],$

where

$$\begin{eqnarray*} R_{11}&=&u_{11}x_{11}+u_{21}x_{12}-v_{11}y_{11}-v_{21}y_{12}\... ... I_{22}&=&v_{12}x_{21}+v_{22}x_{22}+u_{12}y_{21}+u_{22}y_{22}. \end{eqnarray*}$$

See also Real Matrix