A lossless equalized reactive power divider is shown below, along with its dispersion matrix. If port $3$ is terminated in a coupled load, find the noise figure of the splitter and discuss the source of the resulting thermal noise. Assume that the system is at the physical temperature TO$.$

$[S]=\left[\begin{array}{ccc}0& \frac{-j}{\sqrt[\phantom{2}]{2}}& \frac{-j}{\sqrt[\phantom{2}]{2}}\\ \frac{-j}{\sqrt[\phantom{2}]{2}}& \frac{1}{2}& \frac{-1}{2}\\ \frac{-j}{\sqrt[\phantom{2}]{2}}& \frac{-1}{2}& \frac{1}{2}\end{array}\right]$

Figure $1$: Lossless matched reactive power divider with its dispersion matrix.