問題 8-2 の解答例

\begin{equation*}
\tau_{jk}=\sum_\alpha \frac{a_{j\alpha}a_{k\alpha}}{\omega_\alpha}
\end{equation*}

\begin{equation}
\sum_{l=1}^{n} \sum_{m=1}^{n} \tau_{jl}\,\tau_{lm}\,v_{mk} = \delta_{jk}
\tag{8-65}
\end{equation}

\begin{equation}
\tau_{jl}=\sum_{\alpha}\frac{a_{j\alpha}a_{l\alpha}}{\omega_{\alpha}},\quad \tau_{lm}=\sum_{\beta}
\frac{a_{l\beta}a_{m\beta}}{\omega_{\beta}}
\tag{1}
\end{equation}

\begin{align}
&\omega_{\alpha}^{\ 2}\,a_{j\alpha} = \sum_{k=1}^{n} v_{jk}\,a_{k\alpha}= \sum_{k=1}^{n} v_{kj}\,a_{k\alpha}\tag{8-44}\\
&\sum_{j=1}^{n} a_{j\alpha}\,a_{j\beta} = \delta_{\alpha\beta}
\tag{8-48}
\end{align}

\begin{align}
\sum_{l=1}^{n}\sum_{m=1}^{n}\tau_{jl}\tau_{lm}v_{mk}&=\sum_{l}\sum_{m}\left(\sum_{\alpha}\frac{a_{j\alpha}a_{l\alpha}}
{\omega_{\alpha}}\right)\left(\sum_{\beta}\frac{a_{l\beta}a_{m\beta}}{\omega_{\beta}}\right)\,v_{mk}\notag\\
&=\sum_{m}\sum_{\alpha}\sum_{\beta}\frac{a_{j\alpha}a_{m\beta}}{\omega_{\alpha}\omega_{\beta}}
\left(\sum_l a_{l\alpha}a_{l\beta}\right)\,v_{mk}\notag\\
&=\sum_{m}\sum_{\alpha}\sum_{\beta}\frac{a_{j\alpha}a_{m\beta}}{\omega_{\alpha}\omega_{\beta}}
\delta_{\alpha\beta}\,v_{mk}\notag\\
&=\sum_{m}\sum_{\alpha}\left(\sum_{\beta}\frac{a_{j\alpha}a_{m\beta}}{\omega_{\alpha}\omega_{\beta}}
\delta_{\alpha\beta}\right)\,v_{mk}=\sum_{m}\sum_{\alpha}\frac{a_{j\alpha}a_{m\alpha}}{\omega_{\alpha}\omega_{\alpha}}v_{mk}\notag\\
&=\sum_{\alpha}\frac{a_{j\alpha}}{\omega_{\alpha}^{\,2}}\left(\sum_m v_{mk}a_{m\alpha}\right)\notag\\
&=\sum_{\alpha}\frac{a_{j\alpha}}{\omega_{\alpha}^{\,2}}\times\omega_{\alpha}^{\,2}\,a_{k\alpha}\notag\\
&=\sum_{\alpha}a_{j\alpha}a_{k\alpha}
\tag{2}
\end{align}

\begin{align}
\sum_{\alpha} a_{k\alpha}\sum_{j} a_{j\alpha}\,a_{j\beta} &= \sum_{\alpha} a_{k\alpha}\,\delta_{\alpha\beta}\notag\\
\sum_{j}\left(\sum_{\alpha} a_{j\alpha}\,a_{k\alpha}\right)\,a_{j\beta} &= a_{k\beta}
\tag{3}
\end{align}

\begin{equation*}
\sum_{j}\delta_{jk}\,a_{j\beta} = a_{k\beta}
\end{equation*}

\begin{equation}
\sum_{\alpha} a_{j\alpha} \,a_{k\alpha} = \delta_{jk}
\tag{4}
\end{equation}

\begin{equation}
\sum_{l=1}^{n}\sum_{m=1}^{n}\tau_{jl}\tau_{lm}v_{mk} = \sum_{\alpha}a_{j\alpha}a_{k\alpha} = \delta_{jk}
\tag{5}
\end{equation}