(1)

\begin{align*} \left(Q_{\sigma}\right)_{i,j} & =\left(\delta_{i,\sigma\left(j\right)}\right)_{i,j}\\ & =\left(\delta_{j,\sigma\left(i\right)}\right)_{j,i}\\ & =\left(\left(P_{\sigma}\right)_{j,i}\right)_{i,j}\\ & =\left(P_{\sigma}^{T}\right)_{i,j} \end{align*}

(1)-2

\begin{align*} Q_{\sigma} & =\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{1,\sigma\left(2\right)} & \cdots & \delta_{1,\sigma\left(n\right)}\\ \delta_{2,\sigma\left(1\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{2,\sigma\left(n\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{n,\sigma\left(1\right)} & \delta_{n,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)\\ & =\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{2,\sigma\left(1\right)} & \cdots & \delta_{n,\sigma\left(1\right)}\\ \delta_{1,\sigma\left(2\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(2\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{1,\sigma\left(n\right)} & \delta_{2,\sigma\left(n\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)^{T}\\ & =P_{\sigma}^{T} \end{align*}

(2)

\begin{align*} P_{\sigma}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =\sum_{j=1}^{n}\left(\delta_{j,\sigma\left(i\right)}\right)_{i,j}x_{j}\\ & =\left(\sum_{j=1}^{n}\delta_{j,\sigma\left(i\right)}x_{j}\right)_{i,1}\\ & =\left(x_{\sigma\left(i\right)}\right)_{i,1}\\ & =\left(\begin{array}{c} x_{\sigma\left(1\right)}\\ x_{\sigma\left(2\right)}\\ \vdots\\ x_{\sigma\left(n\right)} \end{array}\right) \end{align*} \begin{align*} P_{\sigma}^{T}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =\sum_{j=1}^{n}\left(\delta_{i,\sigma\left(j\right)}\right)_{i,j}x_{j}\\ & =\sum_{j=1}^{n}\left(\delta_{\sigma^{\bullet}\left(i\right),j}\right)_{i,j}x_{j}\\ & =\left(\sum_{j=1}^{n}\delta_{\sigma^{\bullet}\left(i\right),j}x_{j}\right)_{i,1}\\ & =\left(x_{\sigma^{\bullet}\left(i\right)}\right)_{i,1}\\ & =\left(\begin{array}{c} x_{\sigma^{\bullet}\left(1\right)}\\ x_{\sigma^{\bullet}\left(2\right)}\\ \vdots\\ x_{\sigma^{\bullet}\left(n\right)} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{n}\end{array}\right)P_{\sigma} & =\left(P_{\sigma}^{T}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)\right)^{T}\\ & =\left(\begin{array}{c} x_{\sigma^{\bullet}\left(1\right)}\\ x_{\sigma^{\bullet}\left(2\right)}\\ \vdots\\ x_{\sigma^{\bullet}\left(n\right)} \end{array}\right)^{T}\\ & =\left(\begin{array}{cccc} x_{\sigma^{\bullet}\left(1\right)} & x_{\sigma^{\bullet}\left(2\right)} & \cdots & x_{\sigma^{\bullet}\left(n\right)}\end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{n}\end{array}\right)P_{\sigma}^{T} & =\left(P_{\sigma}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)\right)^{T}\\ & =\left(\begin{array}{c} x_{\sigma\left(1\right)}\\ x_{\sigma\left(2\right)}\\ \vdots\\ x_{\sigma\left(n\right)} \end{array}\right)^{T}\\ & =\left(\begin{array}{cccc} x_{\sigma\left(1\right)} & x_{\sigma\left(2\right)} & \cdots & x_{\sigma\left(n\right)}\end{array}\right) \end{align*} \(Q_{\sigma}\)についての証明も同様。

(2)-2

\begin{align*} P_{\sigma}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{2,\sigma\left(1\right)} & \cdots & \delta_{n,\sigma\left(1\right)}\\ \delta_{1,\sigma\left(2\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(2\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{1,\sigma\left(n\right)} & \delta_{2,\sigma\left(n\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)\\ & =\sum_{k=1}^{n}\left(\begin{array}{c} \delta_{k,\sigma\left(1\right)}x_{k}\\ \delta_{k,\sigma\left(2\right)}x_{k}\\ \vdots\\ \delta_{k,\sigma\left(n\right)}x_{k} \end{array}\right)\\ & =\left(\begin{array}{c} x_{\sigma\left(1\right)}\\ x_{\sigma\left(2\right)}\\ \vdots\\ x_{\sigma\left(n\right)} \end{array}\right) \end{align*} \begin{align*} P_{\sigma}^{T}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{2,\sigma\left(1\right)} & \cdots & \delta_{n,\sigma\left(1\right)}\\ \delta_{1,\sigma\left(2\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(2\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{1,\sigma\left(n\right)} & \delta_{2,\sigma\left(n\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)^{T}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)\\ & =\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{1,\sigma\left(2\right)} & \cdots & \delta_{1,\sigma\left(n\right)}\\ \delta_{2,\sigma\left(1\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{2,\sigma\left(n\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{n,\sigma\left(1\right)} & \delta_{n,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right)\\ & =\sum_{k=1}^{n}\left(\begin{array}{c} \delta_{1,\sigma\left(k\right)}x_{k}\\ \delta_{2,\sigma\left(k\right)}x_{k}\\ \vdots\\ \delta_{n,\sigma\left(k\right)}x_{k} \end{array}\right)\\ & =\left(\begin{array}{c} x_{\sigma^{\bullet}\left(1\right)}\\ x_{\sigma^{\bullet}\left(2\right)}\\ \vdots\\ x_{\sigma^{\bullet}\left(n\right)} \end{array}\right) \end{align*}

(3)

任意の列ベクトル\(\left(x_{1},x_{2},\cdots,x_{n}\right)^{T}\)に対し、
\begin{align*} P_{\tau}P_{\sigma}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =P_{\tau}\left(\begin{array}{c} x_{\sigma\left(1\right)}\\ x_{\sigma\left(2\right)}\\ \vdots\\ x_{\sigma\left(n\right)} \end{array}\right)\\ & =\left(\begin{array}{c} x_{\tau\left(\sigma\left(1\right)\right)}\\ x_{\tau\left(\sigma\left(2\right)\right)}\\ \vdots\\ x_{\tau\left(\sigma\left(n\right)\right)} \end{array}\right)\\ & =\left(\begin{array}{c} x_{\tau\circ\sigma\left(1\right)}\\ x_{\tau\circ\sigma\left(2\right)}\\ \vdots\\ x_{\tau\circ\sigma\left(n\right)} \end{array}\right)\\ & =P_{\tau\circ\sigma}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) \end{align*} となるので、\(P_{\tau}P_{\sigma}=P_{\tau\circ\sigma}\)が成り立つ。
これを使って、
\begin{align*} Q_{\tau}Q_{\sigma} & =P_{\tau}^{T}P_{\sigma}^{T}\\ & =\left(P_{\sigma}P_{\tau}\right)^{T}\\ & =P_{\sigma\circ\tau}^{T}\\ & =Q_{\sigma\circ\tau} \end{align*} となるので、\(Q_{\tau}Q_{\sigma}=Q_{\sigma\circ\tau}\)が成り立つ。

(4)

\begin{align*} P_{\sigma}^{T}\left(\begin{array}{c} x_{1}\\ x_{2}\\ \vdots\\ x_{n} \end{array}\right) & =\left(\begin{array}{c} x_{\sigma^{\bullet}\left(1\right)}\\ x_{\sigma^{\bullet}\left(2\right)}\\ \vdots\\ x_{\sigma^{\bullet}\left(n\right)} \end{array}\right)\\ & =P_{\sigma^{\bullet}}\left(\begin{array}{c} x_{\sigma^{\bullet}\left(1\right)}\\ x_{\sigma^{\bullet}\left(2\right)}\\ \vdots\\ x_{\sigma^{\bullet}\left(n\right)} \end{array}\right) \end{align*} \begin{align*} P_{\sigma^{\bullet}}P_{\sigma} & =P_{\sigma^{\bullet}\circ\sigma}\\ & =I \end{align*} \begin{align*} P_{\sigma} & P_{\sigma^{\bullet}}=P_{\sigma\circ\sigma^{\bullet}}\\ & =I \end{align*} となるので、
\[ P_{\sigma^{\bullet}}=P_{\sigma}^{-1} \] となる。
同様に
\begin{align*} Q_{\sigma^{\bullet}} & =Q_{\sigma}^{T}\\ & =Q_{\sigma}^{-1} \end{align*} が成り立つ。

(5)

\begin{align*} \det\left(P_{\sigma}\right) & =\det\left(\begin{array}{cccc} \delta_{1,\sigma\left(1\right)} & \delta_{2,\sigma\left(1\right)} & \cdots & \delta_{n,\sigma\left(1\right)}\\ \delta_{1,\sigma\left(2\right)} & \delta_{2,\sigma\left(2\right)} & \cdots & \delta_{n,\sigma\left(2\right)}\\ \vdots & \vdots & \ddots & \vdots\\ \delta_{1,\sigma\left(n\right)} & \delta_{2,\sigma\left(n\right)} & \cdots & \delta_{n,\sigma\left(n\right)} \end{array}\right)\\ & =\sum_{\tau\in S_{n}}\sgn\left(\tau\right)\left(P_{\sigma}\right)_{1,\tau\left(1\right)}\left(P_{\sigma}\right)_{2,\tau\left(2\right)}\cdots\left(P_{\sigma}\right)_{n,\tau\left(n\right)}\\ & =\sum_{\sigma\in S_{n}}\sgn\left(\sigma\right)\delta_{\tau\left(1\right),\sigma\left(1\right)}\delta_{\tau\left(2\right),\sigma\left(2\right)}\cdots\delta_{\tau\left(n\right),\sigma\left(n\right)}\\ & =\sgn\left(\sigma\right) \end{align*} これを使って、
\begin{align*} \det\left(Q_{\sigma}\right) & =\det\left(P_{\sigma}^{T}\right)\\ & =\det\left(P_{\sigma}\right)\cmt{\because\det\left(A^{T}\right)=\det\left(A\right)}\\ & =\sgn\left(\sigma\right) \end{align*} となる。