一般化散乱行列とインピーダンス行列の関係
インピーダンス行列\([\boldsymbol{Z}]\)より，
\(\boldsymbol{V} = [\boldsymbol{Z}] \boldsymbol{I}\)．
これより\(\boldsymbol{V}\)を消去すると，入射波振幅\(\hat{\boldsymbol{a}}\)，反射波振幅\(\hat{\boldsymbol{b}}\)は，
\begin{eqnarray}
\hat{\boldsymbol{a}} &=& \frac{1}{2}
\left[ \sqrt{\hat{R}} \right]^{-1}
\Big( [\boldsymbol{Z}] + [\hat{Z}] \Big) \boldsymbol{I}
\\
\hat{\boldsymbol{b}} &=& \frac{1}{2}
\left[ \sqrt{\hat{R}} \right]^{-1}
\Big( [\boldsymbol{Z}] - [\hat{Z}]^* \Big) \boldsymbol{I}
\end{eqnarray}
これを，\(\hat{\boldsymbol{b}} = [\hat{\boldsymbol{S}}] \hat{\boldsymbol{a}}\)に代入して，
\begin{gather}
\left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] - [\hat{Z}]^* \Big) \boldsymbol{I}
= [\hat{\boldsymbol{S}}] \left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] + [\hat{Z}] \Big) \boldsymbol{I}
\\
\left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] - [\hat{Z}]^* \Big)
= [\hat{\boldsymbol{S}}] \left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] + [\hat{Z}] \Big)
\tag{1} \label{ZmZSZpZ}
\end{gather}
よって，一般化散乱行列\([\hat{\boldsymbol{S}}]\)は，
\begin{gather}
[\hat{\boldsymbol{S}}]
= \left[ \sqrt{\hat{R}} \right]^{-1}
\Big( [\boldsymbol{Z}] - [\hat{Z}]^* \Big)
\Big( [\boldsymbol{Z}] + [\hat{Z}] \Big) ^{-1}
\left[ \sqrt{\hat{R}} \right]
\label{eq:gSZ}
\end{gather}
逆に，式\eqref{ZmZSZpZ}より，
\begin{gather}
\left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] - [\hat{Z}]^* \Big)
= [\hat{\boldsymbol{S}}] \left[ \sqrt{\hat{R}} \right]^{-1} \Big( [\boldsymbol{Z}] + [\hat{Z}] \Big)
\\
\Big( [U] - [\hat{\boldsymbol{S}}] \Big) \left[ \sqrt{\hat{R}} \right]^{-1} [\boldsymbol{Z}]
=\left[ \sqrt{\hat{R}} \right]^{-1} [\hat{Z}]^*
+ [\hat{\boldsymbol{S}}] \left[ \sqrt{\hat{R}} \right]^{-1} [\hat{Z}]
\nonumber \\ \hspace{37.5mm}
= \left( [\hat{Z}]^* + [\hat{\boldsymbol{S}}] [\hat{Z}] \right) \left[ \sqrt{\hat{R}} \right]^{-1}
\end{gather}
これより，インピーダンス行列\([\boldsymbol{Z}]\)は，
\begin{gather}
[\boldsymbol{Z}] = \left[ \sqrt{\hat{R}} \right] \Big( [U] - [\hat{\boldsymbol{S}}] \Big)^{-1}
\left( [\hat{Z}]^* + [\hat{\boldsymbol{S}}] [\hat{Z}] \right) \left[ \sqrt{\hat{R}} \right]^{-1}
\label{eq:ZgS}% \\ \hspace{9mm}
\end{gather}