Show that the decimation transformation of a one-dimensional Ising model, with \(l=2\), can be written in terms of the transfer matrix \(\boldsymbol{P}\) as

\[
\begin{equation*}
\boldsymbol{P}^{\prime}\left\{\boldsymbol{K}^{\prime}\right\}=\boldsymbol{P}^{2}\{\boldsymbol{K}\} \tag{1}
\end{equation*}
\]

where \(\boldsymbol{K}\) and \(\boldsymbol{K}^{\prime}\) are the coupling constants of the original and the decimated lattice, respectively. Next show that, with \(\boldsymbol{P}\) given by

\[
(\boldsymbol{P}\{\boldsymbol{K}\})=e^{K_{0}}\left(\begin{array}{cc}
e^{K_{1}+K_{2}} & e^{-K_{1}}  \tag{2}\\
e^{-K_{1}} & e^{K_{1}-K_{2}}
\end{array}\right)
\]

see equation (13.2.4), relation (1) leads to the same transformation equations among \(\boldsymbol{K}\) and \(\boldsymbol{K}^{\prime}\) as (14.2.8a, b, and c).